CAPUT XL.
De resolutione syllogismorum ad impossibile et ex hypothesi.
Amplius autem ex hypothesi syllogismos non est tentandum reducere, nam non est ex iis quae posita snnt reducere; non enim por syllogismum ostensi sunt, sed ad placitum concessi sunt omnes. Ut si quis ponat, si una quasdam potestas non sit contrariorum, neque disciplinam esse unam; deinde disputet quoniam non est una potestas contrariorum, utsanativi et ægrotativi, simulenim idem erit sanativum et fegrotativum. Quoniam autem non est omaium contrariorum una potestas,ostensum est, scd quoniam disciplina non una, non est ostensum; quamvis confiteri sit necesse,at non ex syllogismo, vcrum ex hypothesi; hoc igitur non est reducere, quoniam non una potestas est: hic enim fortasse erat syllogismus, illud autem hypothesis. Similiter autem in his qui per impossibile concluduntur,nam neque hoc est resolvere,sed ad impossibile quidem reductio est; syllogismo enim monstratur; alterum autem non est, nam ex hypothesi concluditur. Differunt, aucem a prædictis quoniam in illis quidem oportet prius confiteri, si debet concedere, ut si ostendatur una potestas contrariorum,etdisciplinam esse eamdem; hic autem et non prius confessi concedunt, eo quod manifestum sit falsum, ut posita diametro symetro, eo quod imparia esse æqualia partibus.
CHAPTER XL.
On the resolution of syllogisms to the impossible and from hypothesis.
Furthermore, one must not attempt to reduce syllogisms from hypothesis, for it is not from the things that have been posited that one can reduce them; for they have not been shown by syllogism, but all have been conceded at will. For example, if someone posits that if there is not one certain power of contraries, neither is there one discipline; then let him argue that there is not one power of contraries, as of the healthy and the sick—for at the same time the same thing will be healthy and sick. But that there is not one power of all contraries has been shown, yet that the discipline is not one has not been shown; although it must be confessed, not from syllogism but from hypothesis. This, therefore, is not to reduce, since there is not one power—for here perhaps was the syllogism, but that was the hypothesis. Similarly also in those which are concluded through the impossible, for neither is this to resolve, but it is a reduction to the impossible; for it is shown by syllogism, but the other is not, for it is concluded from hypothesis. They differ, however, from the aforementioned, because in those it is necessary first to confess, if one is to concede, as if it be shown that there is one power of contraries, and that the discipline is the same; but here they concede even without having first confessed, because the false is manifest, as when the diameter is posited as commensurable, because the odd are equal to the even parts.
Plures autem et diversi terminantur ex conditione, quos prospicere oportet, et notare apte. QucC ergo horum differentiæ, et quoties fiunt, qui sunt ex hypothesi, postea dicemus. Nunc autem tantum sit nobis manifestum quoniam non est resolvere in figuras hujusmodi syllogismos, et ob quam causam diximus.
CAPUT XLI.
De reciproqua reductione syllogismorum unius figuræ in aliam.
Quæcunque autem in pluribus figuris monstrantur proposita, si in altera syllogizetur, est reducere syllogismum in alteram, ut eum qui in prima est privativum in secundam figuram, et eum qui in media est in primam. Non omnes autem,3ed quosdam,eritautem in sequentibus manifestum.Si enim a nulli b, autem omni c, a nulli c, sic ergo prima figura; si autem convertatur privativa, media erit. Nam a quidem nulli, autem omni inerit.Similiter autem et si non universalis, sed particularis fit syllogismus,ut si a quidem nulli b,b autem alicui c, conversaenimprivativamediaeritfigura.Eorum au.. temsyllogismorum,qui suntin secunda figura, universales quidem reducentur in primam figuram, particularium autem alter solum.
But many and diverse terms are delimited by condition, which one must foresee and note aptly. What, then, are the differences of these, and how many times they occur, and which are from hypothesis, we will say afterwards. For now, let it only be clear to us that it is not possible to resolve syllogisms of this kind into the figures, and for what cause we have said.
CHAPTER XLI.
On the reciprocal reduction of syllogisms of one figure into another.
Whatever propositions are demonstrated in several figures, if a syllogism is made in one of them, it is possible to reduce the syllogism into the other, as the privative one that is in the first figure into the second figure, and the one that is in the middle into the first. Not all, however, but some; and this will be made clear in what follows. For if a belongs to no b, but to every c, then a belongs to no c; thus, therefore, the first figure; but if the privative proposition is converted, it will be the middle figure. For a will belong to no b, but to every c. Similarly also if the syllogism is not universal but particular, as if a belongs to no b, but b to some c; for when the privative proposition is converted, it will be the middle figure. But of those syllogisms that are in the second figure, the universal ones are reduced into the first figure, but of the particular ones only one.
Insit enim a quidem nulli, vero omni,conversa privativa prima erit figura, nam quidem nulli a, a autem omni inerit.Si autem prsedicativum quidemsitad b,privativumautem ad c, primus terminus ponendus est c, hoc enim nulli a, a -autem.omni b, quare nulli inerit c, ergo et nullic, convertitur enim privativa. Si autem particularis sit syllogismus,quando privativum quidem erit ad majorem extremitatem,resolvetur in primam figuram, ut si a nulli b, autem alicui c, conversa enim privativa prima erit figura, nam quidem nulli a, a autem alicuic. Quando vero prcedicativum, non resolvetur, ut si a quidem omni b, vero non omni, non enim suscipit conversionem a b, neque cum fit, erit syllogismns. Rursus qui in tertia quidem sunt figura, non resolvuntur omnes in primam,qui autem sunt in prima,omnes in tertiam. Insit enim a quidem omni b, autem alicui c, ergo quia convertitur particularis praedicativa, inerit et alicui b,a vero omni inerat, quare fit tertia figura, Et si privativus sit syllogismus,similiter: convertitur enim particularis affirmativa, quare a quidem nulli 6,cautem alicui inerit. Eorum autem syllogismorum quisunt in postrema figura unus tantum non resolvitur in primam, quando non universalis ponitur privativa,alii autem omnes resolvuntur.
For let a belong to none, but to every, the converted privative will be the first figure; for a belongs to none, but a will belong to every. But if the predicative term is to b, and the privative to c, the first term must be posited as c; for this belongs to no a, but a to every b, wherefore it will belong to no c, therefore also to no c, for the privative converts. But if the syllogism is particular, when the privative is to the major extreme, it will be resolved into the first figure, as if a belongs to no b, but to some c; for the converted privative will be the first figure, for a belongs to none, but a to some c. But when it is predicative, it will not be resolved, as if a belongs to every b, but not to every c; for a b does not admit of conversion, nor when it is made will there be a syllogism. Again, those in the third figure are not all resolved into the first, but those in the first are all resolved into the third. For let a belong to every b, but to some c; therefore, because the particular predicative converts, it will also belong to some b, but a belonged to every; wherefore the third figure is made. And if the syllogism is privative, similarly; for the particular affirmative converts, wherefore a will belong to no b, but c to some. But of the syllogisms that are in the last figure, one alone is not resolved into the first, when the privative is not posited as universal, but all the others are resolved.
Praedicentur enim de omni c., et a et 6, ergo convertetur ad utrumque particulariter; inerit ergo a alicui b,quare erit prima figura, siquidem a omni c, vero alicui 6; et si a quidem omni c, autem alicui c,eadem ratio, convenitur enim ad bc.Si autem 6quidem omni c, a autem alicui c. primus ponendus b, nam omni c, autem alicui a,quare alicui a, quoniam autem convertilur particularis,et a alicui inerit.Et si privativus sit syllogism us universalibus terminis,similiter sumendum. Insit enim omni c, a autem nulli c, ergo alicui inerit c, a autem nulli c,quare erit medium c. Similiter autem et si privativa quidem si universalis, prædicativa autem particularis, nam a quidem nulli c, c, autem alicui inerit. Si autem particularis sumatur privaliva,non erit resolutio,ut si quidem omni c, a autem alicui non inest, conversa enim c,utræque propositiones eruntparticulares.
For let both a and b be predicated of every c; therefore it will be converted to each particularly; therefore a will belong to some b, so that it will be the first figure, since a belongs to every c, but to some b. And if a belongs to every c, but to some c, the same reasoning holds, for it is brought to bc. But if b belongs to every c, and a to some c, b must be placed first; for it belongs to every c, but to some a, therefore to some a; and since the particular is converted, a will also belong to some. And if the syllogism is privative with universal terms, it must be taken similarly. For let it belong to every c, but a to no c; therefore it will belong to some c, but a to no c, so that c will be the middle term. Similarly also if the privative is universal, but the predicative particular; for a belongs to no c, but c will belong to some. But if the privative is taken as particular, there will be no resolution, as if it belongs to every c, but a does not belong to some; for when c is converted, both propositions will be particular.
Manifestum autem quoniam ad resolvendum ad se invicem figuras, quæ ad minorem extremitatem est propositio, convertenda in utrisque figuris, hac conversa,transitio fit; eorum autem qui in media sunt figura, alter quidem resolvitur, alter vero non resolvitur in tertiam, nam cum sit universalis privativa, resolvitur.Sienim a nulli quidem b,alicui autem c, utraque similiter convertitur ad a,quare quidem nulli a, vero alicui,medium ergo a.Quando autem a omni b, autem alicui non insit, non fit resolutio, neutra enim propositionum ex conversione universalis. Qui autem ex tertia sunt figura, resolventur in mediam, quando fuerit universalis privativa, ut si a nulli c, autem alicui, aut omni nam c, a quidem nulli. autem alicui inerit.Si autem particularis sit privativa, non resolvetur, non enim suscipit conversionem particularis negativa. Manifestum ergo quoniam iidem syllogismi non resolvuntur in his figuris, qui nec in primam resolvebantur, et quoniam in primam figuram reductis syllogismis, isti soli syllogismi per impossibile clauduntur.Quomodo ergo oportet syllogismos reducere,et quoniam resolvuntur figuræ in se invicem, manifestum ex, dictis.
It is evident that for resolving figures into one another, the proposition that pertains to the lesser extreme must be converted in both figures, and when it has been converted, the transition is made. Of those in the middle figure, one is resolved and the other is not resolved into the third; for when the universal privative proposition is present, it is resolved. For if a belongs to no b, but to some c, both are converted in like manner to a; therefore a belongs to no a, but to some a; therefore the middle term is a. But when a belongs to every b, but does not belong to some c, no resolution occurs, for neither proposition is universal by conversion. Those in the third figure are resolved into the middle when the universal privative proposition is present, as if a belongs to no c, but to some or to every c; for c belongs to no a, but to some a. But if the privative proposition is particular, it will not be resolved, for a particular negative does not admit conversion. It is evident, therefore, that the same syllogisms are not resolved in these figures which were not resolved into the first figure, and that when syllogisms have been reduced to the first figure, these alone are concluded through impossibility. In what way, then, one ought to reduce syllogisms, and that the figures are resolved into one another, is evident from what has been said.
CAPUT XLII.
De syllogismis infinitis, et regulis consequentiarum.
DifFert autem in construendo vel destruendo opinari, aut idem, aut diversum significare, non esse hoc, et esse non hoc, ut non esse album, ei quod est esse non album; non enim idem significant,nec est negatio ejus puse est esse album ea quae est esse non album,sed non esse album. Ratio autem hujus hæc est; similiter enim se habet possibile est ambulare ad possibile non ambulare, id quae est esse album ad esse non album, et scit bonum ad scit non bonum: nam scit bonum velsciens bonum nihil differt, neque potest ambulare vel est potens acubulare; quare est opposita, non potest ambulare et non est potens ambulare. Si igitur non est potens ambulare idem significat et est potens non ambulare, ipsa simul inerunteidem,nam idem potestambulareetnonambulare, et idem sciens bonum et non bonum est. Affirmatio autem et negatio non sunt oppositas simul in eodem.
CHAPTER XLII.
On infinite syllogisms, and the rules of consequences.
But in constructing or destroying an argument, it makes a difference whether one holds that "not to be this" and "to be not this" signify the same or different things—for example, "not to be white" and "to be not white." For they do not signify the same thing, nor is "not to be white" the negation of "to be white"; rather, "to be not white" is. The reason for this is as follows: for "it is possible to walk" stands in the same relation to "it is possible not to walk" as "to be white" does to "to be not white," and "he knows the good" to "he knows the not-good." For "he knows the good" differs in no way from "he is knowing the good," nor "he is able to walk" from "he is capable of walking"; wherefore the opposites are "he is not able to walk" and "he is not capable of walking." If, then, "he is not able to walk" signifies the same as "he is able not to walk," both will be present in the same subject at once; for the same person can walk and not walk, and the same person knows the good and the not-good. But affirmation and negation are not opposites present at the same time in the same subject.
Quemadmodum ergo non idem est, non scire bonum et scire non bonum, nec esse non bonum et non esse bonum, idem, nam proportionalium, si alterum sit, et alterum,nec esse noncequale et non esse aequale idem, huic enim quod est non æquale subjacet aliquid,et hoc estinsequale,illi vero nibil, eo quod æquale quidemvelinsequale non omne est, aequale autem vel non æquale omne: amplius, est non album lignum et non est album lignum non simul sunt, si enim est lignum non album, erit Iignum,quod autem non est album lignum, non necesse est esse lignum: quare manifestum est quoniam non est ejus quod est bonum, est non bonum, negatio; si ergo de omni uno vel affirmatio, vel negatio vera, si non est negatio, palam quoniam affirmatio aliquo modo erit; affirmationis autem omnis, negatio est, et hujus ergo, ea quae est non est, non bonum.Habent autem ordinem hunc ad invicem, sit esse quidem bonum in quo a, non esse autem bonum in quo b, esse autem non bonum in quo sub b, non esse autem non bonum in quo sub a, omni ergo inerit aut a, aut b, et nulli eidem, et omni aut c, aut d, et nulli eidem, et cui inest, necesse est omni inesse.
Therefore, just as "not to know the good" and "to know the not-good" are not the same, neither are "to be not-good" and "not to be good" the same; for in proportional pairs, if one holds, the other does too, nor are "to be not-equal" and "not to be equal" the same. For under "not equal" something is subordinated, and this is "unequal," whereas under the other there is nothing, because not everything is either equal or unequal, but everything is either equal or not equal. Moreover, "the wood is not white" and "the wood is not-white" are not simultaneous; for if the wood is not-white, there will be wood, but what is "not white wood" need not be wood. Hence it is clear that the negation of "good" is not "not-good." Therefore, if of every single thing either the affirmation or the negation is true, and it is not the negation, it is plain that the affirmation will in some way hold; but of every affirmation there is a negation, and therefore of this one too, namely the one that is "is not good." Now these have this order among themselves: let "to be good" be that in which a, "not to be good" that in which b, "to be not-good" that which is under b, "not to be not-good" that which is under a. Therefore a or b will belong to everything, and neither to the same thing, and c or d to everything, and neither to the same thing, and whatever c belongs to, it is necessary that it belong to everything.
Si enim verum est dicere quoniam est non album, et quoniam non est album, verum; impossibile est enim simul esse album et esse non album,aut esse lignum album et esse lignum non album: quare si non affirmatio,etnegatio inerit.Ei autem quod est b, non semper c, quod enim omnino non est lignum, neque lignum erit album, nec non album. E converso autem cui inest a, et omni inest, aut enim c, aut: quoniam autem non possunt simu: esse non album et ese album, inerit, nam de eo quod est album verum est dicere quoniam non est non album. De autem non omnino a erit, nam de eo quod omnino non est lignum, non verum est dicere a quoniam est lignum album; quare verum est, et a non verum, quoniam est lignum album.
Palam autem quoniamet a et nulli eidem insunt, sed et contingit eidem alicuiinesse.Similiter autem sehabent et privationes ad prædicationes eadem positione: sit enim a quale in quo a, non aequale in quod b, inæquale in quo c, non ineequale in quo d. In pluribus autem quorum his quidem inest, illis vero non inest idem, negatio quidem similiter vera fit,ut quoniam non sunt alba omnia., aut quoniam non est album unumquodque, aut quoniam est non album unumquodque, aut quoniam omnia sunt non alba, falsum est. Similiter autem et ejus quæ est omne animal album, non hæc (est non album omne animal) negatio, ambae enim falsæ, sed es,non omne animal album.
For if it is true to say that it is not-white, and that it is not white, [it is] true; for it is impossible for the same thing to be white and to be not-white at the same time, or for there to be white wood and not-white wood; wherefore if there is no affirmation, the negation will be present. But for that which is b, c does not always [follow]; for that which is altogether not wood will be neither white wood nor not-white. Conversely, however, for that to which a belongs, [c] also belongs to every [instance]; for either c, or—since it is impossible for not-white and white to be present at the same time—[the negation] will be present; for concerning that which is white it is true to say that it is not not-white. But concerning that which is not altogether [wood], a will not hold; for concerning that which is altogether not wood, it is not true to say a, that it is white wood; wherefore it is true, and a is not true, that it is white wood.
It is clear, moreover, that both a and [its negation] are not present in the same thing, but it also happens that they are present in some same thing. In like manner, privations are related to predications in the same arrangement; for let a be "equal," in which [is] a; let "not equal" be that in which [is] b; let "unequal" be that in which [is] c; let "not unequal" be that in which [is] d. But in many things of which some have [a property] and others do not have the same, the negation likewise becomes true, as that all are not white, or that each one is not white, or that each one is not-white, or that all are not-white; [but] it is false. In like manner also, of the [proposition] "every animal is white," this is not the negation: "every animal is not-white" (for both are false), but rather: "not every animal is white.
Quoniam autem palam quod aliud significat est non album, et non est album, et illa. quidem affirmatio, haec vero negatio, manifestum quoniam non est idem modus monstrandi utrumque, ut quoniam quidquid est animal, non est album,aut contingit non esse album, et quoniam verum dicere non album,hoc enim est esse non album. Sed verum quidem dicere, est album, sive non album, idem modus.Constructive enimambae per primam ostenduntur figuram, nam verum ei quod est similiter ordinatur, ejus enim quae est, verum dicere album, non hæc, verum dicere non album, negatio,sedhæc, non est verum dicere album. Si enim verum dicere quidquid est homo musicum esse,autnonmusicum esse,quidquid est animal sumendum musicum esse, aut non musicum esse, et ostensum est. Non esse autem musicum quidquid esthomo,destructive monstratursecundum dictostresmodos.Simpliciter autem quando sicse habent a et b, ut simul quidem eidem non contingant, omni autem de necessitilte alterum, et rursum et similiter. Sequitur autem id quod est c, a, et non convertitur, et id quod est sequetur d, et non convertitur, et a quidem et contingunt eidem, autem et non contingunt.Primum ergo quoniam id quod est sequitur d, hinc manifestum quoniam eorum quse suntc alterum ex necessitate omniinest, cui autem non contingitc, eo quod simul infert a, a autem et non contingunt eidem, manifestum quoniam sequetur b.
But since it is clear that "is not-white" signifies something different from "is not white"—the former being an affirmation, the latter a negation—it is manifest that the manner of demonstrating each is not the same: for instance, "whatever is an animal is not white," or "it is possible not to be white," and "it is true to say not-white" (for this is to be not-white). But "it is true to say [it is] white" and "not-white" [are demonstrated] in the same manner. For both are shown constructively through the first figure; for "true" is ordered similarly to the thing, since the negation of "it is true to say white" is not "it is true to say not-white," but "it is not true to say white." For if it is true to say that whatever is a man is musical or not musical, one must take that whatever is an animal is musical or not musical, and it has been demonstrated. But that whatever is a man is not musical is demonstrated destructively according to the three modes stated. Simply, however, when a and b are so related that they cannot at the same time belong to the same thing, but by necessity one of them belongs to every thing, and again similarly. And it follows that c [belongs to] a, and it does not convert, and that which is d will follow, and it does not convert, and a and [c] do belong to the same thing, but [b and d] do not belong. First, then, that which is [c] follows d—from this it is manifest that one of the things that are c belongs by necessity to every thing to which c cannot belong; and since a is inferred together with it, and a and [b] cannot belong to the same thing, it is manifest that b will follow.
Rursum quoniam ei quod est a non convertitur c, omni autem vel c, vel d, contingit a, et eidem inesse; autem et £ non contingit, eo quod consequitur a id quod est c, accidit enim quiddam impossibile.Manifestum est ergo quoniam nec ei quod est convertiur, eo quod contingit simul a, inesse. Accidit autem aliquoties in hujusmodi terminorum ordine falli,eo quod opposita non sumantur recte, quorum necesse est omni alterum inesse: ut se a et non contingunt simul eidem,necesse est auteminessecui non alterum, alterum, et rursus et rl similiter, cui autem omni sequitur a,accidet enim cui d,b inesse ex necessitate, quod falsum est; si sumatur enim negatio eorum quae sunta b, ea quae est in quibus f, et rursus eorum quae sunt d, ea quæ est in quibus g. Necesse est igitur omni inesse vel a, vel f, aut enim af'firmationem,aut negationem,et rursum, aut c, aut; affirmatio enim et negatio, et cui omni a subjacet, quare cui omni boc quod est g. Rursum quoniam eorum quæ sunt omni alterum, eteorum quae sunt similiter. Sequitur autem id quod est f, et id quod est sequitur 6, hoc enim seimus.
Again, since c is not convertible with a, but it is possible for a to belong to every c or d and to the same thing; and it is not possible for both to belong, because c follows from a, something impossible results. It is clear, therefore, that neither is it convertible with that which is, because it happens that a belongs at the same time. But sometimes error occurs in an ordering of terms of this kind, because opposites are not correctly assumed, of which it is necessary that one belong to every thing: for example, if a and not-a cannot belong at the same time to the same thing, it is necessary that one belong to whatever the other does not, and again similarly for the rest. But whatever a follows in every case, it will happen that b belongs by necessity to that to which d belongs, which is false. For if one takes the negation of the things that are a and b, namely that which is in the things in which f is, and again of the things that are d, that which is in the things in which g is. It is necessary, therefore, that either a or f belong to every thing; for it is either an affirmation or a negation, and again either c or not-c; for it is an affirmation and a negation, and whatever a is subject to in every case, wherefore whatever g is. Again, since of the things that are, one of two opposites belongs to every thing, and similarly of the things that are not. But f follows, and that which is follows b; for this we know.
Si ergo a id quod est c, et id quod est sequetur b, boc autem falsum; e contrario enim erat in his (quæ sic se babent) consequentia, Non enim fortasse necessariumomni inesse, auta aut /, nee (aut:nonenimest negatio ejus quodestnhocquodestf,nam boni non bonumnegatio;nonautem est idem hoc quod estnonbonum ei quod est nequebonum neque non bonum; similiter autem et in d, nam negationes qllæ sumptæ sunt, duæ sunt.
LIBER SECUNDUS.
CAPUT PRIMUM.
In quot ergo figuris, et per quales, et quot propositiones, et quando, et quomodo fit syllogismus, ampliusautem ad quæ perspiciendum construenti et destruenti, et quomodo oporteat quærere de proposito secundum unamque artem, amplius autem per quam viam sumemus, quæ in singulis sunl principia jam pertransivimus. Quoniam autem alii quidem syllogismorum sunt universales, alii vero particulares: universales quidem omnes semper plura syllogizant,particularium autem praedicativi quidem plura, negativi vero conclusionem solam. Nam alix quidem propositiones convertuntur, privativa vero non convertitur. Conclusio vero aliquid de aliquo est, quare alii quidem syllogismi plurasyllogizant: ut si a ostensum sit omni aut alicui ineEse, et alicui a necessarium est inesse,et si nulli hinesse a, etb nulli n, bocautem aliud est a priore. Si autem a alicui non insit, non necesse est et alicuia non inesse; contingit enim omni a inesse.
If, then, a is the same as c, and what is the same as c will also follow b, but this is false; for the converse was the consequence in things disposed in this way. For perhaps it is not necessary that it belong to every one, or a or f, nor is it the negation of that which is this which is f; for the negation of good is not good; but this which is not good is not the same as that which is neither good nor not good; and similarly also in d, for the negations that have been assumed are two.
BOOK TWO
CHAPTER ONE
In how many figures, then, and through what kinds, and how many propositions, and when, and in what way a syllogism comes about, and further what things one must look to in constructing and destroying, and in what way one ought to inquire about a proposed question according to each art, and further by what path we shall take the principles that are in each case—these things we have already gone through. Since, however, some syllogisms are universal and others particular: all universal syllogisms always conclude more things, while of particular syllogisms the affirmative conclude more, but the negative only the conclusion. For some propositions are convertible, but the privative one is not convertible. The conclusion, however, is something about something; wherefore some syllogisms conclude more things: for instance, if a has been shown to belong to every one or to some one, and it is necessary that a belong to some one, and if a belongs to no one of these, then b belongs to no one of these, but this is something other than the previous one. If, however, a does not belong to some one, it is not necessary that a also not belong to some one; for it is possible for it to belong to every one.
Haec ergo communis omnium causa universalium et part.icularium. Est autem de llniversalibus, et aliter dicere, quæcumqueenim aut sub medio autsubconclusione sunt, omnium erit idem syllogismus, si illa quidem in medio, baec vero in conclusione ponantur, ut si a conclusio per n, quaecunque sub baut sub sunt, necesse est de omnibus dicia, namd sim toto b, et in a, et erit in a. Rursum si e in toto c, et in toto a, et e in loto a erit. Similiter autem et si privativus sit syllogismus. In secunda aut figura quod sub conclusione erit, solum erit syllogizare, ut si a insit nulli b, et omni c, conclusio quoniam nulli inest; si autem sub est, manifestum quoniam non inestei 6, iisautem quoe sunt sub a,quoniam non inest, non palam est persyllogisrnum, et si non inest quod est e, si est e sub a, sed inesse.quidem nulli per syllogismum ostensum est, non inesse veroa hoc quod estb, indemonstratum sumpt.um est, quare nec per syllogismum accidit non inessec.In particularibus autem,eorum quidem quæ sub conclusione sunt, non erit necessarium, Non enim fit syllogismus, quando ea sumpta fuerit particularis; eorum autem quae sunt sub medio, erit ornnium,verumtamen non per syIlogismun, utsiaomni h, et alicui c: nam ejus quod sub est positum, non erit syllogismus, ejus vero quod sub erit, sed non proper eum qui prius factus est syllogismum.
This, then, is the common cause of all universals and particulars. But concerning universals, it can also be stated otherwise: for whatever things are either under the middle term or under the conclusion, the same syllogism will apply to all of them, if those are placed in the middle term and these in the conclusion—for example, if a is the conclusion through n, whatever things are under b or under c, it is necessary that they be said of all; for if d is in the whole of b, and b in a, then d will also be in a. Again, if e is in the whole of c, and c in the whole of a, then e will also be in the whole of a. The same holds similarly if the syllogism is privative. In the second figure, however, only what is under the conclusion will be syllogized—for example, if a belongs to no b and to every c, the conclusion is that it belongs to no c; but if something is under c, it is evident that it does not belong to it, yet for those things that are under a, that it does not belong to them is not made clear through the syllogism—and if that which is e does not belong, if e is under a, but it was assumed to belong. That it belongs to none has been shown through the syllogism, but that it does not belong to this thing which is b was assumed undemonstrated; therefore it does not result through the syllogism that it does not belong. In particular cases, however, of those things that are under the conclusion, it will not be necessary; for no syllogism arises when a particular has been assumed. But of those things that are under the middle term, there will be a syllogism for all—yet not through the syllogism, for example, if a belongs to every b and to some c; for of that which is placed under c there will be no syllogism, but of that which is under b there will be, though not on account of the syllogism that was made previously.
Similiter autem et in aliisfiguris, nam ejus quidem quod sub conclusione est non erit, alterius vero erit, verum non per syllogismum, eo quod et in universalibus ex indemonstrata propositione quæ sunt sub medio ostendebantur; quare neque hic erit, vel et in illis.
CAPUT II.
Quoll ex falsis in prima figura uerum colligatur.
Est ergo sic se babere, ut veræsint propositiones per quasft syllogismus; est autem ut falsae, estvero ut hæc quidem vera, illa autem falsa, concl usio autem aut vera, aut falsaex necessitate. Ex veris ergo non est falsum syIlogizare, ex falsis autem verum, tamen non propter quid,sed quia, nam ejus qui est propterquid non est ex falsis syllogism us, ob quam autem causam insequentibusdicetur,Primumergo quoniam ex veris non possibiIe falsum syllogizare hinc manifestum. Si enim cum est a, necesse est esse b, si non est b, necesse est a non esse; si ergo verum est a,necesse est et verum esse,aut accidet idem simul et esse etnonesse, hoc autemimpossi. bile. Non autemquoniamponitura unusterminus, accipiatur, contingere uno aliquo existente, ex nece"Jsitate aliquid accidere, non enim potest. Nam quod accidit ex necessit:tte conclusio est, per quæ autem fit ad minimum tres sunt termini, duo autem intervalla et propositiones.
Similarly also in the other figures; for it will not be of the one that is under the conclusion, but it will be of the other, yet not through a syllogism, because even in universals the things that are under the middle term were shown from an indemonstrable proposition; wherefore neither will it be here, or even in those.
CHAPTER II.
That a true conclusion is inferred from false premises in the first figure.
It is, then, possible for the premises through which the syllogism is formed to be true; it is also possible for them to be false; and it is possible for one of them to be true and the other false, while the conclusion is either true or false by necessity. From true premises, then, it is not possible to syllogize a false conclusion; but from false premises a true conclusion is possible, yet not as to the reason why, but as to the fact; for the syllogism that gives the reason why does not come from false premises. The cause of this, however, will be stated in what follows. First, then, that it is not possible to syllogize a false conclusion from true premises is manifest from this: for if, when a is, it is necessary that b be, then if b is not, it is necessary that a not be. If, therefore, a is true, it is necessary that b also be true, or else it will happen that the same thing at once is and is not, which is impossible. Nor is it because one term is posited that it should be accepted that it is possible, when some one thing exists, for something to come about by necessity; for this cannot be. For what comes about by necessity is the conclusion, and the terms through which it comes about are at least three, with two intervals and premises.
Si ergo verum est cui omni inest et a, cui autem et b, cui c, necesse est a inesse, et non potest hoc falsum esse, simul enim erit idemet non inerit; ergo a ut unum, positum est duas propositiones colIigere. Similiterautem se habet et in privativis, non enim est ex veris ostendere falsum. Ex falsis autem est verum syllogizare, utrisque propositionibus faIsis,et una; hac autem non utralibet contingit, sed secunda, si quidem totam sumamus falsam, non tota autem sumpta est utralibet. Insit enim a onni c, ei autem quod est nulli, nec insit Cj conlingit autem boc,ut nulli lapidi unimal, et lapis nulli homini; si igitur sumatur a omni b,etb omni C, a omnic inerit, quare exutrisque falsis vera est conclusio, omnis enim homo animal.SirniIiter autem et privativum:in?itenim nulli, nec a, nee b, a autem omni, ut si eisdem terminis sumptis medium ponatur homo, lapidi enim nec animal, nea homo nulli inest, homini autem omni animal; quare si cuiquidem omni inest, sumamus nulli inesse, cui vero non inest, omni inesse, ex falsis utrisque vera eritconclusio. Similiter autem oslendeturet si in aliquo utraque falsa sumatur. Si autem altera ponaturfalsa, prima quidem tota falsa existente, ut a b, non erit conclusio vera, autem erit. Dico autem totam falsam quod contrariam veræ, ut si quod nulli in est,omni sumptum est; aut si quod omni nulli inesse.
If therefore it is true that a belongs to everything to which b belongs, and b to everything to which c belongs, then a must belong to c, and this cannot be false; for at the same time it will and will not belong to the same thing. Therefore a, as one term, has been posited to combine two propositions. The same holds also in privative syllogisms, for it is not possible to prove a falsehood from true premises. But from false premises it is possible to syllogize a truth, when both propositions are false, and when one is false; and this does not happen with either one indifferently, but with the second, if indeed we take the whole proposition as false, whereas when it is not taken as a whole, either one may be false. For let a belong to every c, and to that which is b let it belong to none, nor let c belong to b; this can happen, as animal belongs to no stone, and stone to no man. If therefore it is assumed that a belongs to every b, and b to every c, then a will belong to every c; hence from both false premises the conclusion is true, for every man is an animal. Similarly also in the privative case: let b belong to no c, and neither a nor b belong to any c, but let a belong to every b; as, if with the same terms assumed, man is posited as the middle term, for animal belongs to no stone, nor does man belong to any stone, but animal belongs to every man. Therefore if we assume that what belongs to every one belongs to none, and that what does not belong belongs to every one, from both false premises the conclusion will be true. Similarly it will also be shown if both are assumed false in a particular proposition. But if one is posited as false, when the first is wholly false, as a belongs to b, the conclusion will not be true, but it will be true when the other is false. I call a proposition wholly false when it is the contrary of the true one, as if what belongs to none is assumed to belong to every one, or what belongs to every one to belong to none.
Insit enim a nulli, autem omni; si ergo quidem propositionem su mamus veram, a autem falsam totam, et omni inesse a, impossible est a conclusionem veram esse, nulli enim inerat a eorum quæ sunt c,si quidem cui nulli, autem omni c. Similiter autem nec si a omni inest, et omni c, sumpta sit autem quidem vera propositio, a autem falsa tota, et nulli cui inest a, conclusio falsa erit, omni enim inest a, siquidem cui omni et a,b autem omni c. Manifestum ergo quoniam prima tota sumpta fal sa, sive affirmativa, sive privativa altera autem vera, non fit vera conclusio. Non tota autem sumpta falsa erit: nam si a quidem omni inest, autem alicui, autem omni c, ut animal, cygno quidem omni, albo autem alicui, album autem omni cygno, si sumatur a omni b, et omni c, a omni inerit vere, omnis enim cygnus animal. Similiter autem et si privativa ani-sit a; possibile est enim ab quidem alicui inesse, vero nulli, autem omni c, ut mal alicui albo, nivi vero nulli, album vero omni nivi; si ergo sumatur a quidem nulli b, autem omni c, a nulli inerit.
For let a belong to none, but b to all; if therefore we take the proposition b as true, but a as wholly false, and [say] that a belongs to all, it is impossible for the conclusion to be true; for a belonged to none of the things that are c, if indeed [it belongs] to none of those to which b belongs, but b to all c. Similarly, neither if a belongs to all, and to all c, but the proposition b is taken as true, while a is wholly false, and [we say] that a belongs to none of those to which it belongs, the conclusion will be false; for a belongs to all, if indeed [it belongs] to all to which b belongs, and b to all c. It is clear, therefore, that when the first [premise] is taken as wholly false, whether it is the affirmative or the privative one, while the other is true, the conclusion does not come out true. But when it is not taken as wholly false, it will be [true]: for if a belongs to all b, and to some c, but b to all c, as animal to all swan, but to some white, and white to all swan, if a is taken to belong to all b, and to all c, a will truly belong to all; for every swan is an animal. Similarly also if a is privative; for it is possible for a to belong to some b, but to none of those to which b belongs, and b to all c, as malice to some white, but to no snow, and white to all snow; if therefore a is taken to belong to none of b, but to all c, a will belong to none.
Si autem a quidem propositio tota sumatur veru, autem tota falsa, erit syllogismus verus, nihil enim prohibet a, et b,et omni inesse,b autem nulli c, ut quæcunque ejusdem generis sunt species non subalternæ, nam animal et homini et equo inest, equus autem nulli homini inest.; si ergo sumatur a omnib, et omni c, con ciusio vera eri, tota falsa propositione. Similiter autem cum universalis privativa est a propositio, contingit enim a neque b, neque nulli inesse, et nulli c, ut ex alio genere speciebus diversum genus, nam animal nec musicas, nec medicinæ inest, neque musica medicinæ. Sumpta ergo a quidem nulli b, autem omni c, vera erit conclusio. Et si non tota laIsa sit c, sed in aliquo, etiam sic erit conclusio vera. Nihil enim prohibet a, et betc toti inesse, 6autem alicui c, utgenus speci ei et differentiæ, nam animal homini omni et omni gressibili, homo autem alicui gressibili, et non omni; si ergo a omni b, et omni sumatur, a omni inerit, quod quidem erat verurn.SimiIiter autem cum pri-, vativa est a, propositio, conlingit enim a nec 6, nec nulli inesse, vero alicui c, at genus ex alio genere speciei et differentiæ,nam animal nec sapientiae nulli inest, nec contemplationi, sapientia vero alicui contemplationi; si ergo sumatur a nulli b, autem omni c, nulli inerit a, hoc autem erat verum.
But if the proposition a is taken as wholly true, while b is wholly false, the syllogism will be true; for nothing prevents a from belonging both to b and to every c, while b belongs to no c—as, for instance, whatever species of the same genus are not subordinate to one another; for animal belongs both to man and to horse, but horse belongs to no man. If, therefore, a is taken as belonging to all b and to all c, the conclusion will be true, though the proposition b is wholly false. Similarly also when the universal privative proposition is a; for it is possible for a to belong neither to b nor to any c, as a genus different from another genus is related to the species of that other genus; for animal belongs neither to music nor to medicine, nor does music belong to medicine. If, therefore, a is taken as belonging to no b, but to all c, the conclusion will be true. And if c is not taken as wholly false, but as false in some part, even so the conclusion will be true. For nothing prevents a from belonging to all b and all c, while b belongs to some c—as a genus to a species and a differentia; for animal belongs to every man and to every walking thing, but man belongs to some walking things, and not to all. If, therefore, a is taken as belonging to all b and to all c, a will belong to all c, which was indeed true. Similarly also when a is a privative proposition; for it is possible for a to belong neither to b nor to any c, but to some c—as a genus from another genus is related to a species and a differentia; for animal belongs neither to wisdom nor to contemplation, but wisdom belongs to some contemplation. If, therefore, a is taken as belonging to no b, but to all c, a will belong to no c, and this was true.
In particularibus autem syllogismis contingit, prima propositione tota falsa existente, altera autem vera, veram esse conclusionem, et a in aliquo Falsa existente, autem vera, et a quidem vera, particulari autem falsa, et utrisque existentibus falsis.Nibil enim prohibet a quidem nulli inesse, autem alicui, et alicmc inesse, ut animal nulli nivi, albo autem alicui inest, et nix albo alicui. Si ergo ponatur medium nix primurn antem animal, et sumatur a quidem toti inesse, autem alicui c, a tota falsa, autem vera, et conclusio vera. Similiter autem et cum privativa est a propositio, possibile est enim a quidem toti inesse, autem alicui non inesse, vero alicui inesse, ut animal homini quidem omniinest,album autem aliquod non sequitur, homovero alicui alboinest; quare si medio posito homine sumatur a nulliinesse, et a'icui c, vera fit conclusio, cum sit tota falsaa proposito. Et si in aliquo sit falsa a propositio, vera existente,eritconcIusio vera. Nihil enim probibet a, et b, et c, alicui inesse, autem alicui c, ut animaIalicui puIchro,et alicui magno, et pulchrum alicui magno inest; si ergo sumatur a omni b,etb' alicui c, et a quidempropositioin aIiquo falsaerit, autem vera, et conclusio vera. Similiter autem et cum privativa est a propositio, nam iidem erunt termini, et:simiIiterpositi ad demonstrationem. Rursum si a quidem vera, autem falsa, vera erit, conclusio.
But in particular syllogisms it happens that when the first proposition is wholly false and the other true, the conclusion is true; and when a belongs to some and is false, while the other is true; and when a is true but the particular proposition false; and when both are false. For nothing prevents a from belonging to none, while b belongs to some, and b belongs to some c—as, for instance, animal belongs to no snow, but white belongs to some, and snow belongs to some white. If, then, snow is posited as the middle term and animal as the first, and a is taken to belong to the whole of b, and b to some of c, then a is wholly false, b is true, and the conclusion is true. Likewise also when the proposition a is privative; for it is possible that a belongs to the whole, while b belongs to some and does not belong, and b belongs to some—as animal belongs to every man, but white does not follow some [man], and man belongs to some white. Hence if man is posited as the middle term and a is taken to belong to no b, and b to some c, the conclusion comes out true, although the proposition a is wholly false. And if the proposition a is false in some respect, while b is true, the conclusion will be true. For nothing prevents a, b, and c from belonging to some, and b from belonging to some c—as animal belongs to some beautiful thing, and to some great thing, and the beautiful belongs to some great thing. If, then, a is taken to belong to every b, and b to some c, then the proposition a will be false in some respect, b will be true, and the conclusion will be true. Likewise also when the proposition a is privative; for the same terms will be, and similarly posited, for the demonstration. Again, if a is true and b false, the conclusion will be true.
Nihil enim prohibet a quidem toti inesse 6, autem alicui, et nulli inesse: ut animal'. cygnus quidem omni,nigro autem alicui,cygnus vero nulli nigro; quare sisumatur a omni b, et alicui c, vera erit conclusio, cum sit falsa c. Similiter autem et privativasumpta a propositione, possibile enima quidem nulIi, autem alicui non inesse, et nulli r,, ut genus ex alio genere speciei et accidenti ejus speciebus,nam animal quem numero nulli inest, alio vero non alicui, numerus autem nulli albo; si ergo medium ponatur numerus,et sumatur a quidem nulli b,b autem alicui c, a alicui non inerit, quod fuit verum, cum a quidem si propositiovera, autem falsa.Etsi in aliquo sitfalsa a 6, falsa autem et c, erit conclusio vera. Nihil enim prohibet a alicui bet alicui inesse utrique,b autem nulli c, ut si sit contrarium ipis c,etamto accidentia eidem generi, nam animal alicui albo et alicui nigro inest, album autem nulli nigro inest; si ergo sumatur a omni b, et alicui c, vera erit conclusio. Et privativaquidem sumpta a b, similiter. Nam idem termini, etsimililer ponenturad demonstrationem. Et ex utrisque falsis eritconclusio vera. Possibile est enim a quidem nulli, caulem alicui inesse, vero nulli c. Ut genus ex alio genere speciei, et accidenti speciebus ejus, animal enim numero quidem nulli, albo vero aIicuiinest, et numerusnuIliaIbo.
For nothing prevents a from belonging to the whole of b, or to some, and to none: as animal; for swan belongs to every [animal], but to some black thing, while swan belongs to no black thing; therefore if a is taken as belonging to every b, and to some c, the conclusion will be true, though it is false of c. And similarly when the proposition a is taken as privative; for it is possible for a to belong to none, or to some, and to none of r, as a genus from another genus of a species and an accident of its species; for animal belongs to no number, but to some other thing, while number belongs to no white thing; therefore if number is posited as the middle term, and a is taken as belonging to no b, but b to some c, a will not belong to some c, which was true, though the proposition a was true and the other false. And if a is false of some b, and c is also false, the conclusion will be true. For nothing prevents a from belonging to some b and to some of both, while b belongs to no c, as if it were the contrary of c itself, and also accidents of the same genus; for animal belongs to some white thing and to some black thing, but white belongs to no black thing; therefore if a is taken as belonging to every b, and to some c, the conclusion will be true. And similarly when a b is taken as privative. For the same terms, and similarly posited, will serve for the demonstration. And from both false premises the conclusion will be true. For it is possible for a to belong to no c, but to some other thing, and for b to belong to no c. As a genus from another genus of a species, and an accident of its species; for animal belongs to no number, but to some white thing, and number to no white thing.
Si ergo sumaturaomni b, et balicui j, conclusio quidem vera, propositiones vero ambae falsæ. Similiter autem etcum privativa esta b. Nihil enim prohibet a quidem toti inesse,c autem alicui non inesse, etneque nulIi c, ut animal cygno quidem omni, nigro autem alicui non inest, cygnus vero nulli nigro:quare si sumatur a nulli b,b autem alicui ca alicuic non inerit; ergo conclusio quidem vera, propositiones autem falsae.
CAPUT III.
Quod colligatur verum ex falsis ita secunda figura,
In media autem figura omnino contingit per falsa verum syIlogizare,et utrisque propositionibus totis falsis sumptis, et bac quidem vera, illa tota falsa, utralibet falsa posita,et si utraque in aliquo falsa?, et si haec quidem simpliciter vera, illa autem in aliquo falsa, et in universalibus, et in particularibus syllogismis. nulli,Si enim a quidem nulli inest, autem omni, ut lapidi animal quidem homini autem omni, si contrariae ponantur propositiones, et si sumatur a quidem omni, vero nulli,ex falsis totis propositionibus erit vera conciusio. Similiter autem et si a inest quidem omni, vero nulli, nam idem erit syllogismus. Rursum si altera quidem tota falsa, altera autem tota vera. Nihil enim prohibet a et et omni inesse,b autem nulli c, ut genus non subalternis speciebus.
If then a is taken as belonging to every b, and b to some c, the conclusion is indeed true, but both premises are false. It is the same also when the privative proposition is about b. For nothing prevents a from belonging to the whole of c, but not to some c, and to no c: as animal belongs to every swan, but not to some black thing, while swan belongs to no black thing. Therefore if a is taken as belonging to no b, and b to some c, a will not belong to some c; therefore the conclusion is indeed true, but the premises are false.
CHAPTER III.
That a true conclusion is inferred from false premises in this way in the second figure.
In the middle figure it is altogether possible to syllogize a true conclusion through false premises, both when both premises are taken wholly false, and when one is true and the other wholly false, whichever of the two is posited as false, and if both are false in some respect, and if one is simply true and the other false in some respect, both in universal and in particular syllogisms. For if a belongs to no c, but to every b, as animal to no stone but to every man, if the contrary propositions are posited, and if a is taken as belonging to every c but to no b, the conclusion will be true from wholly false premises. It is the same also if a belongs to every c but to no b; for the syllogism will be the same. Again, if one premise is wholly false and the other wholly true. For nothing prevents a from belonging both to every c and to no b, as a genus to species that are not subordinate.
Nam animal equo omni, et homini inest, et nullus homo equus;si ergo sunaaturanimal buicquidem omni,ilIi vero nulli inesse,bæc quidem erit falsa,iIla vero tota vero, et conclusio vera,ad quodlibet posito privativo. Et si altera in aliquo f'alsa, altera autem tota vera, possibile est enim a quidem alicui in esse, autem omni, et nulli c, animal albo quidem alieui, corvo autem omni, album vero nulli corvo. Si ergo sumatur a quidem nulli, autem toti inesse, a quidem propositio in aliquo falsa est, a autem tota vera, et conclusio vera, et transposita quidem privativa,sirniIiter.Nam per eosdem termi. nos demonstratio.Et si affirmativa quidcm propoaitio in aliquo faIsa,privativa autem tota vera,. nihil enim prohibet a quidem alicui inesse, autem toti non inesse, et nulli c, ut animal albo quidem aIicui,pici autem nulli,album vero nulli pici: quare si sumatur,a toti inesse,c autem nulli,a quidem in aliquo falsa, a r: autem tota vera, et conclusio vera. Et si utrasque propositiones in aliquo falsae, erit conclusio vera, possibile est enim a, et b, et alicui inesse, autem nulli c, ut animal, et albo alicui, et nigro alicui, album vero nulli nigro. Si ergo sumatur a quidem omni,c autem nulli,ambae quidem propositiones in aliquo falsæ,conclusio autem vera; similiter autem transposita privativa per terminos.
For animal belongs to every horse and to every man, and no man is a horse. If then it is taken that animal belongs to every one of these and to none of those, the former premise will be false, the latter wholly true, and the conclusion true, with the privative proposition posited in relation to anything. And if one of the premises is false in some respect, while the other is wholly true—for it is possible for a to belong to some and to belong to every, and to no c, as animal belongs to some white thing and to every raven, while white belongs to no raven. If then it is taken that a belongs to none and to every, the premise about a is false in some respect, while the other is wholly true, and the conclusion is true; and when the privative proposition is transposed, similarly. For the demonstration is through the same terms. And if the affirmative premise is false in some respect, while the privative premise is wholly true—for nothing prevents a from belonging to some and not belonging to every, and to no c, as animal belongs to some white thing and to no woodpecker, while white belongs to no woodpecker; wherefore if it is taken that a belongs to every and c to none, the premise about a is false in some respect, while the other is wholly true, and the conclusion is true. And if both premises are false in some respect, the conclusion will be true; for it is possible for a and b to belong to some and to no c, as animal belongs to some white thing and to some black thing, while white belongs to no black thing. If then it is taken that a belongs to every and c to none, both premises are false in some respect, but the conclusion is true; and similarly when the privative proposition is transposed, through the terms.
Manifestum autem et in particularibus syllogism is, nibil enim prohibet a quidem omni,c autem alicui inesse,et alicui non inesse,ut animal omni homini, albo autem alicui, homo vero alicui albo non inerit. Si ergo ponatur a quidem nulli inesse,c autem alicui inesse, universalis quidem propositio tota falsa, particularis autem vera, et conclusio vera. Similiter autem et affirmativa sumpta a b, possibile est enim a quidem nulli, autem alicui non inesse,et alicui non inesse,ut animal nulli inanimato,aIbo autem alicui,et inanimatum non inerit alicui albo. Si ergo ponatur a quidem omni, vero alicui non inesse,a quidem propositio universalis tota faIsa,a autem vera, et conclusio vera. Et universali quidem vera posita, minori autem particulari falsa,nihil enim prohibet a nec nec nullum sequi, et alicui non inesse, ut animal nulli numero nec inanimato,et numerus aliquod inanimatum non sequitur. Si ergo ponatur a quidem nulli,c autem alicui,et conclusio vera, et universalis proposito vera, particularisautem falsa.Affirmativa autem universali similiter posita, possibile est enim a et et toti inesse, autem aliquod non sequi, ut genus speciem et differentiam.
It is also evident in particular syllogisms; for nothing prevents A from belonging to every C and to some C and not to some C, as animal belongs to every man, but to some white thing, while man will not belong to some white thing. If then it is assumed that A belongs to no C, but to some C, the universal proposition is wholly false, the particular is true, and the conclusion is true. Similarly also if the affirmative premise AB is assumed; for it is possible that A belongs to no C and not to some C and not to some C, as animal belongs to no inanimate thing, but to some white thing, and the inanimate thing will not belong to some white thing. If then it is assumed that A belongs to every C, but not to some C, the universal proposition A is wholly false, but the other is true, and the conclusion is true. And when the universal is assumed true, but the particular minor premise false; for nothing prevents A from following neither no C nor some C, and from not belonging to some C, as animal follows no number nor inanimate thing, and number does not follow some inanimate thing. If then it is assumed that A belongs to no C, but to some C, the conclusion is true, and the universal proposition is true, but the particular is false. Similarly also when the universal affirmative is assumed; for it is possible that A belongs to the whole and does not follow some, as genus to species and difference.
Nam animal omnem hominem et totum gressibile sequitur, homo vero non omne gressibile: qua re si sumatur a quidem toti inesse, autem alicui non inesse,universaIis quidem propositio vera,particularis faIsa,concIusio autem vera.Manifesturn autem quoniam et utrisque falsis erit conclusio vera, siquidem contingit a et et huic quidem omni,illi vero nulli inesse, vero aliquod non sequi,nam sumpto a quidem nulli,:a autem alicui inesse, propositiones quidem ambæ falsæ, conclusio autem vera. Similiter autem et cum prædicativa fuerit uni versalis propositio, particularis autem privativa, possibile est enim a quidem nullum, autem omne sequi,et alicui non inesse, ut animal disciplinam quidem nullam, hominem autem omnem sequitur, disciplina vero non omnem hominem. Si ergo su matur a quidem toti inesse, autem aliquod non sequi,propositiones quidem falsæ,conclusio autem vera.
For animal follows every man and every whole thing that walks, but man does not follow every thing that walks. Therefore, if it is assumed that A belongs to every C but not to some C, the universal proposition is true, the particular false, and the conclusion true. It is also clear that the conclusion will be true even when both propositions are false, since it is possible for A to belong to every C and to no C, and for B not to follow some C. For if A is taken to belong to no C, but to some C, both propositions are false, yet the conclusion is true. Likewise also when the proposition is universal affirmative but the particular is privative; for it is possible for A to follow no B but every C, and not to belong to some C, as animal follows no discipline but every man, while discipline does not follow every man. If, then, it is assumed that A belongs to every C but does not follow some C, the propositions are false, but the conclusion is true.
CAPUT IV.
Quod ex falsis verum identidem colligatur in tertiafigura.
Erit autem et in postrema figura per falsas totas, et in aliquo utraque, et altera quidem vera, altera autem falsa, et hæc quidem in aliquo falsa, illa autem tota vera,et e converso, et quotquot modis aliter possibile est transumere propositiones. Nihil enim prohibet nec a nec nulli inesse, a autem alicui inesse, ut nec homo, nec gressibile, nullum inanimatum sequitur,homo autem alicui gressibili inest; si ergo sumatur a et omni messe,propositiones quidem totæ falsæ conclusio autem vera. Similiter autem ei cum hæc quidem est privativa, illa vero affirmativa, possibile est enim quidem nulli inesse,a autem omni, et a alicui non inesse, ut nigrum nulli cygno, animal autem omni, et ani mal non omni nigro; quare si sumatur quidem omni c, a vero nulli, alicui a non inerit, et conclusio quidem vera, propositiones autem falsæ. Et, si in aliquo fuerit utraque falsa, erit conclusio vera, nihil enim prohibet et a et alicui inesse, et a alicui b, ut album et pulchrum alicui animali inest, et album alicui pulchro; si ergo ponatur a et omni inesse, propositiones quidem in aliquo falsæ, conclusio autem vera. Et privativa a posita, inesse,b vero alicui inesse, et a non omni inesse, ut album alicui animali non inesse.
CHAPTER IV.
That a true conclusion is sometimes inferred from false premises in the third figure.
It will also happen in the last figure through wholly false premises, and in the case of particular premises both false, and one true and the other false, and the one false in a particular, the other wholly true, and conversely, and in as many other ways as it is possible to transpose the propositions. For nothing prevents neither A nor B from belonging to any C, while A belongs to some B: for example, neither man nor walking belongs to any inanimate thing, but man belongs to some walking thing. If, then, it is assumed that A belongs to every B, the premises are wholly false but the conclusion true. Likewise when one of them is privative and the other affirmative: for it is possible for B to belong to no C and A to every C, and for A not to belong to some B: for example, black belongs to no swan, but animal to every swan, and animal not to every black thing. Therefore, if it is assumed that B belongs to every C and A to no C, A will not belong to some B, and the conclusion is true while the premises are false. And if both are false in a particular, the conclusion will be true: for nothing prevents both A and B from belonging to some C, and A from belonging to some B: for example, white and beautiful belong to some animal, and white to some beautiful thing. If, then, it is posited that A belongs to every B, the premises are false in a particular but the conclusion is true. And with A posited as privative, that it belongs, and B as belonging to some C, and A as not belonging to every B: for example, white does not belong to some animal.
Pulchrum autem alicui:nest,et album non omni pulchro: quare si sumatur a quidem nulli, autem omni, utraque propositiones quidem in aliquo faIsæ, concIusio au. tem vera. Similiter autem et hæc quidem tota falsa, illa vero tota vera sumpta.PossibiIe est enim a et omne sequi,et a alicui non inesse,ut animal et album omne cygnum sequitur,et animal non omni inest albo; positis igitur bis terminis,si sumatur quidem toti inesse,a vero toti non inesse,h quidem tota erit vera, a autem tota faIsa,et conclusio vera. Similiter autem et. si quidem falsa, a autem vera,nam hi quidem termini ad demonstrationem, nigrum, animatum, cygnus. Sed et si utraeque assumantur affirmare, nihil enim prohibet quidem omne sequi,a autem toti non inesse, et a alicui inesse,ut omni cygno animal,nigrum vero nulli cygno,et nigrum inest alicui animali: quare si sumatur a et omni inesse,b quidem tota vera, a autem tota falsa, et conclusio vera. Similiter autem et a sumpta vera,nam per eosdem terminos demonstratio. Rursum hac quidem Lota vera esistente,illa vero in aliquo l'aIsa,possibile est enim quidem omni inesse,a autem alicui et alicui b,ut bipes quidem omni hommi,pulchrum non omni,et pulchrum alicui bipedi inest.Si ergo sumatur a et toti inesse,b quidem tota vera, a autem in aliquo faIsa,concIusio autem vera.
But it is possible for something to be beautiful and for white not to belong to every beautiful thing; therefore if one premise is taken as belonging to none and the other as belonging to all, both propositions are in some respect false, but the conclusion is true. And similarly, one premise may be taken as wholly false and the other as wholly true. For it is possible for something to follow every instance of a thing, and yet for that thing not to belong to some instance of another, as animal follows every swan, and animal does not belong to every white thing. So when the terms are set out in this way, if one premise is taken as belonging to all and the other as belonging to none, the one will be wholly true, the other wholly false, and the conclusion true. And similarly if one is false and the other true; for the terms for this demonstration are black, animate, swan. But also if both premises are assumed affirmative—for nothing prevents one term from following every instance of another, while the second does not belong to every instance of the first and yet belongs to some instance of it, as animal belongs to every swan, black to no swan, and black belongs to some animal. Therefore if one premise is taken as belonging to all and the other as belonging to none, the one will be wholly true, the other wholly false, and the conclusion true. And similarly if the one is taken as true; for the demonstration is through the same terms. Again, when one premise is wholly true and the other false in some respect—for it is possible for one term to belong to every instance of another, while the second belongs to some instances of the first and to some instances of the second, as two-footed belongs to every man, beautiful not to every man, and beautiful belongs to some two-footed thing. If, then, one premise is taken as belonging to all and the other as belonging to none, the one will be wholly true, the other false in some respect, but the conclusion true.
Similiter autem et a quidem vera,b autem falsa in aliquo sumpta, transpositis enim eisdem terminis erit demonstratio. Et cum haec quidem est privativa,illa vero affirmativa,quoniam possibile est quidem toti inesse, a autem alicui c,et quando sic se habeant,non omni inesse a.Si ergo assumatur quidem toti inesse, a autem nulli, privativa quidem aliquo falsa, altera autem tota vera,et conclusio erit vera,Rursum quoniam ostensum est quod cum a quidem nulli inest,et alicui, evenit a alicui non inesse, rnanifestum igitur quoniam et cum a tota est vera, autem in aliquo falsa, contingit conclusionem esse veram: si enim sumatur a quidem nulli c, autem omni, a e quidem tota vera, autem in aliquo falsa.Manifestum autem et in particularibus syllogismis quoniam omnino per falsaerit verum, non iidem termini sumendi,et quando universales fuerint propositiones, in prædicativis quidem prædicativi,in privativis autem privativi; nihil enim iffert,cu nulli inerat, universaIiter sumere inesse, et si alicui inerat,universaIiter sumere ad terminorum positionem; similiter autem et in privati vis. Manifestum igitur quod quando sit conclusio falsa, necesse est ea exquibus est oratio faIsaeese,automnia,aut aliqua; quando autem vera,non necesse est verum esse nec aliquod quidem, nec omne.
Likewise also when one premise is true and the other false in some respect; for with the same terms transposed, the demonstration will hold. And when one is privative and the other affirmative, since it is possible for it to belong to the whole and for A to belong to some C, and when they are so related, for A not to belong to every. If then it is assumed that it belongs to the whole, but A to none, the privative premise is false in some respect, while the other is wholly true, and the conclusion will be true. Again, since it has been shown that when A belongs to none and to some, it results that A does not belong to some, it is therefore evident that when A is wholly true but the other false in some respect, it happens that the conclusion is true; for if it is assumed that A belongs to no C, but the other to every, A is wholly true, but the other false in some respect. It is evident also in particular syllogisms that a true conclusion may come about entirely through false premises, but the same terms are not to be assumed, and when the propositions are universal, in affirmative syllogisms the terms must be affirmative, in privative ones privative; for it makes no difference, when it belonged to none, to assume universally that it belongs, and if it belonged to some, to assume universally with respect to the positing of the terms; similarly also in privative syllogisms. It is evident therefore that when the conclusion is false, it is necessary that the premises from which the argument is formed be false, either all or some; but when it is true, it is not necessary that any of them be true, nor all.
Sed est cum nullum sit verum eorum quae sunt in syllogismis, et conclusionem similiter esse veram, non tamen ex necessitate. Causa autem quoniam cum duo sic se habent ad invicern,ut cum alterum sit, ex necessitate esse alterum, hoc cum non sit quidem, nec alterum erit; cum autem sit, non necesse est esse alterum idem autem cum sit, et non sit,impossibile ex necessitate esse idem.Dico autem, cum sit a album, esse magnum ex necessitate, ecum non sit a aIbum,b esse magnum exnecessitate; quando enim cum hoc sit (ut a) album, illud necesse est (ut b) esse magnum, cum autem sit rnagnum,c non esse album, necesse est,si a sit album,c non esse aIbum.Et quando duobus existentibus, cum alterum sit, necesse est alterum esse, hoc autem cum non sit,necesse est a non esse,cum ergo non sit magnum, a non potest album esse, cum vero a non sit album, necesse est magnum esse,accidit ex necessitate cum magnum non sit; idem esse magnum: hoc autem im possibiIe, nam si non est magnum,a non eritalbum ex necessitate; si ergo cum non sita aIbum,b erit magnurn,accidit, si non est magnum,b esse magnurn,ut ertria.
But it happens that when none of the things that are in the syllogisms is true, the conclusion is likewise true, yet not of necessity. The reason is that when two things are so related to each other that, when the one is, the other is of necessity, then if this one is not, neither will the other be; but when it is, there is no necessity for the other to be. And the same thing, when it is and is not, cannot of necessity be the same. I mean: when a is white, it is of necessity large; and when a is not white, b is of necessity large. For when, with this being (say a) white, that (say b) is of necessity large, and when it is large, c is not white, then if a is white, c is of necessity not white. And when, two things existing, with the one being, the other is of necessity, but this one when it is not, a is of necessity not, then when it is not large, a cannot be white; but when a is not white, it is of necessity large. It happens of necessity, when it is not large, that the same thing is large; but this is impossible. For if it is not large, a will of necessity not be white; if therefore, when a is not white, b will be large, it happens that, if it is not large, b is large, as has been said.
CAPUT V.
De circulari ostensione in prima figura.
Circulo autem, et ex se invicem ostendere est per concIusionem,et e converso praedicationem atteram sumentem propositionem concludere rcliquam, quam sumpserat in altero syllogismo, ut si oportuit ostendere quoniam a inest omni c, ostendat autem per c,rursus si monstret quoniam a inest b, sumens a quidem inesse c,c autem,et a inerit b,prius autem econverso sumpsitb in esse c,aut si quoniam inest c,oportert ostendere si sumat a de c, quæ fuit concIusio,b autem dea esse,prius autem sumptum est e converso a de b. AIiLer vero non est ex se invicem ostendere,sive enim aliud medium surmetur, non circuIo,nil enim sumitur eorumdem,sivehorum quiddam,necesee est alterum soIum,nam si ambo, eadem erit conclusio, at oportet diversam esse. In iis igitur quae non convertuntur ex indemonstrata altera propositione fit syllogisrnus,non enim est demonstrare per hos terminos,quoniam medio inest tertiurn,aut primo medium.
CHAPTER V.
On circular demonstration in the first figure.
To demonstrate circularly and from each other is to conclude, by means of the conclusion and conversely, the remaining proposition by taking the other predication, which had been assumed in the other syllogism; for example, if it was required to show that A belongs to every C, and one shows it through C, then again if one shows that A belongs to B, assuming that A belongs to C and C to B, A will also belong to B; but previously the converse was assumed, that B belongs to C. Or if it is required to show that A belongs to C, one assumes A of C, which was the conclusion, and B of A; but previously the converse was assumed, A of B. In another way it is not possible to demonstrate from each other; for whether another middle term is assumed, there is no circle, since nothing of the same terms is assumed; or if something of these is assumed, it is necessary that only one be assumed, for if both, the conclusion will be the same, but it ought to be different. In those things, therefore, which are not convertible, the syllogism comes about from one indemonstrable proposition; for it is not possible to demonstrate through these terms that the third belongs to the middle, or the middle to the first.
In iis autem quae convertuntur,erit omnia monstrare per se invicem, ut si a, et b, etcconvertuntur sibi invicem: ostendatur enim a per medium b, et rursum a per conclusionem,et per propositionem conversam; similiter autem et c, et per conclusionern, et per a propositionem conversam; oportet autem et b, et a propositionem demonstrare, nam his demonstratis asi sumus soIis.Si ergo sumatur omni inesse,etc omnia a, syllogismus erit ejus quod est ad a.Rursus si sumatur,' omni a inesse,et a omni necesse est inesse omni b. In utrisque ergo syltogismis a propositiosumpta estindemonstrata, am aliae probatæ erant: quare si hanc ostenderimus, omnes erunt approbatæ per se invicem; si ergo sumatur comni b,et omni a inesse, utraeque propositiones Jemonstralæ sumuntur. et necesse est inesse a. Manifestum est ergo quoniam in solis iis quae convertuntur,circulo et per se invicem contingit fieri demonstrationes, in aliis vero quemadmodum prius diximus. Accidit autem et in iis eodem quod monstratur uti ad demonstrationem, nam de 6, et de a monstratur sumpto de a dici, autem de aper hasostenditur propositiones; quare conclusione utimur ad de,nonstrationem.
In those things, however, which are convertible, it will be possible to demonstrate all things through one another, as if a and b and c are converted with one another; for let a be demonstrated through the middle term b, and again a through the conclusion and through the converted proposition; and similarly also c, both through the conclusion and through the converted proposition of a. But it is necessary also to demonstrate b and the proposition of a, for when these have been demonstrated we are in possession of all of them. If, therefore, it is assumed that [b] belongs to every [c], and [c] to every a, the syllogism will be of that which is to a. Again, if it is assumed that [b] belongs to every a, and a necessarily belongs to every b. In both syllogisms, therefore, the proposition of a has been assumed undemonstrated, while the others had been proved; wherefore if we have demonstrated this one, all will have been approved through one another. If, therefore, it is assumed that [b] belongs to every c and to every a, both propositions are assumed as demonstrated, and it is necessary that a belongs. It is manifest, therefore, that only in those things which are convertible do demonstrations come about circularly and through one another, but in others in the way we said before. It happens, however, also in these that the same thing which is being demonstrated is used for the demonstration; for it is demonstrated of b and of a when it is assumed that [b] is said of a; but it is shown that [b] is said of a through these propositions; wherefore we use the conclusion for the demonstration.
In privativis autem syllogismis hoc modo monstratur ex se invicem: sit quidem omni ineGse,a autem nulli b, conclusio autem quoniam a nulli c,Si ergo rursum oporteat concludere quoniam a nulli b, quod prius sumptum erat,erit a quidem nulli c, autem omni b, sic enim e converso propositio. Si autem quoniam inesl c,oporteat concludere,non jam similiter convertendum a b, nam eadem propositio est nulli a,et a nulli inesse,eed sumendum,cui a nulli inest, huic omni inesse. Sit enim a nulli inesse, quod quidem fuit conclusio,cui autem a nulli b. si sumatur omni inesse,necesse est ergo omni e inesse: quare cum sint tria,unurnquodque conclusio est facta, et circulo demonstrare, hoc est concIusionern sumentem et e converso alleram propositionem,reliquam syllogizare. In particuiaribus autem syllogismis universalem quidem propositionem non est demonstrare per alias,particularern autem est; quoniam autem non est demonstrare universalem, manifestum, nam universale monstratur per universalia,conclusio autem non est universalis,oportet autem ostendere ex conclusione et altera propositione. Amplius, omnino non fit syllogismus conversa propositione,narn particulares fiunt utræque propositiones.
In privative syllogisms, however, this is shown from one another in this way: let a belong to every c, but a to no b, and the conclusion be that a belongs to no c. If, then, it should again be concluded that a belongs to no b, which was previously assumed, there will be a belonging to no c, but c to every b; for thus the proposition is converted. But if it should be concluded that c belongs, then a to b must no longer be converted in the same way; for the same proposition is "a belongs to no b" and "b belongs to no a," and one must assume that to which a belongs to no b, to this belongs to every b. For let a belong to no c, which indeed was the conclusion, and let a belong to no b. If it is assumed that c belongs to every b, it is necessary, therefore, that c belongs to every b; wherefore, since there are three terms, each one has been made a conclusion, and to demonstrate in a circle is this: taking the conclusion and converting the other proposition, to syllogize the remaining one. In particular syllogisms, however, it is not possible to demonstrate the universal proposition through the others, but the particular one it is possible; that the universal one cannot be demonstrated is clear, for the universal is shown through universals, but the conclusion is not universal, and one must show it from the conclusion and the other proposition. Moreover, a syllogism does not come about at all when the proposition is converted, for both propositions become particular.
Particulare autem est ostendaturenim a de aliquo per b, si ergo sumatur omni a, et conclusio maneat, alicui inerit, fit enim prima figura,et est a rnedium.Si autem fit privativus syHogismus,universalem quidem propositionem non est ostendere,propter boc quod prius dictum est, particularem(si simpliciter convertatur a quemadmodum etin universaltbus) non est, per assumptionem autem est,ut cui a alicui nun insit, alicui inesse; nam aliter se habentibus non fitsyllogismus,eo quod negativa est particularis propositio.
CAPUT VI.
De eadem cyclica ostensione in secunda figura,
In secunda autem figura affirmativam quidem non est ostendere per hunc modum, privativam autem est; ergo praedicativa quidem non ostenditur, eo quod non sunt utræque propositiones affirmativae, nam conclusio privativa,prædicativa autem ex utrisque ostendebatur affirmativis. Privativa autem sic ostenditur: insit enim a omni b,c autem nulIi,conclusia quoniam nulli c,si ergosumaturbomniain- ] esse,et nulli c,necesse est nulli inesse,tlt enim secunda figura, medium 6. Si autem a privativa sumpta sit, altera vero praedicativa, prima erit iigura,narn quidem omni a,b autem nulli c,quare nulli a,ergo nec a b,medium c; ergo per conelusionem quidem et unam propositionem non fitsyIlogismus,assumpta autem altera erit.
But the particular is shown thus: let a belong to some thing through b. If, then, a is taken as belonging to every [b], and the conclusion remains, it will belong to some; for the first figure comes about, and a is the middle term. But if a privative syllogism is made, the universal proposition indeed cannot be shown, on account of what was said before; the particular cannot be shown (if a is converted simply, just as also in universal propositions), but it can be shown by assumption, as that to which a does not belong to some, belongs to some; for with things otherwise disposed no syllogism comes about, because the particular proposition is negative.
CHAPTER VI.
On the same circular demonstration in the second figure.
In the second figure, however, the affirmative cannot be shown in this way, but the privative can. Therefore the predicative is not shown, because both propositions are not affirmative; for the conclusion is privative, whereas the predicative was shown from both affirmatives. But the privative is shown thus: let a belong to every b, but c to no b, and the conclusion be that a belongs to no c. If, then, b is taken as belonging to every a and to no c, it is necessary that it belong to no [c]; for thus the second figure comes about, and b is the middle term. But if a is taken as privative and the other as predicative, there will be the first figure; for a belongs to every b, and b to no c, wherefore a to no c; therefore neither does a belong to b; the middle term is c. Therefore through the conclusion and one proposition no syllogism comes about, but when the other is assumed, it will.
Si autem non universalis sit syllogismus,quæ in toto quidem est propositio non ostenditur,propter eamdem causam quam quidem diximus et prius,quæ autem in parte, ostenditur quando universalis sit prædicativa.Insit einm a omni to, autem non omni, conclusio c; si ergo sumatur omni a, autem non omni,conclusio a alicui non inerit medium b. Si autem est universalis privativa, non ostenditur a propositio, conversa a b, accidit enim utrasque aut alteram propositionem fieri negativam: quare non eritsylIogismus;sed similiter ostendetur quemadmodum et universalibus, si sumatur, cui alicui non inest a alicui inespe.
CAPUT VII.
De cyclica ratiocinatione in tertia MUl'a.
In tertia autem figura, quando utræque propositiones universaliter sumentur, non contingit ostendere per se invicem propositionem. Nam universalis quidem ostenditur per universalia, in hac autem conc:usio semper est particularis: quaremanifestum quonam omnino non contingit ostendere per hanc figuram universalem propositionem. Si autem haec quidem universalis sit, illa vero particularis, quandoque quidem erit, quandoque vero non inerit; quando ergo utraeque prædicativæ sumantur, et universalis sit ad minorem extremitatem, erit; quando vero ad alteram, non erit. Insit enim a omni c, autem alicui c, conclusio a b.
But if the syllogism is not universal, the proposition that is in the whole is not shown, for the same reason that we have also stated before; but the proposition that is in part is shown when the universal proposition is affirmative. For let a belong to every t, but not to every u, and let the conclusion be c; if, then, a is taken as belonging to every [t], but not to every [u], the conclusion will be that a does not belong to some b. But if the universal proposition is privative, the proposition a is not shown by converting a b; for it happens that both propositions, or one of them, become negative; therefore there will not be a syllogism; but it will be shown in the same way as in the case of universal propositions, if it is taken that to which a does not belong to some, [a belongs] to some.
CHAPTER VII.
On circular reasoning in the third figure.
In the third figure, when both propositions are taken universally, it is not possible to show one proposition through the other. For a universal proposition is shown through universals, but in this figure the conclusion is always particular; therefore it is clear that it is altogether impossible to show a universal proposition through this figure. But if one proposition is universal and the other particular, sometimes it will be possible and sometimes not; for when both are taken as affirmative and the universal is attached to the lesser extreme, it will be possible; but when it is attached to the other, it will not. For let a belong to every c, but to some c, and let the conclusion be a b.
Si ergo sumatur omni a inesse conversa universali, et a inesse b, quod erat conclusio, quidem ostensum est alicui inesse, autem alicui tJ, non est ostensum, quamvis necesse est si alicui b, et alicui inesse; sed non idem est hoc iili, et illud huic inesse, sed assumendum est, si hoc alicui illi, etalt-erum alicui huic,hoc autem sumptojam non sit ex conclusione et altera propositione syllogismus. Si autem quidem omni c, a autem alicui c, erit ostendere a c, quando sumatur quidem omni binesse, a autem alicui; nam sic omni binest,a autem alicui b, necesse est a alicui inesse, medium 6. Et cum fuerit haec prædicativa quidem, illa vero privativa,universalis autem prædicativa, ostendetur altera. Insit enimbomnic, aautem aIicuinoninsit, conclusio quoniam a alicui non inest. Si ergo assumatur omni inesse, inerat auteII1 et a non omni b, necesse est a alicui non inesse medium b. Cum autem privativa universalis sit, non ostenditur altera nisi sicut in prioribus, si sumatur cui hoc alicui non inest, alterum alicuHnesse, ut si a nulli c, autem alicui, conclusio quoniam a alicui non inest. Si ergo sumatur cui ft alicui non inest,eidem alicui inesse, necesse est alicui L inesse; aliter autem non est convertentem universalem propositionem ostendere alteram, nullo enim modo erit syllogismus.
If, then, it is assumed that A belongs to every C by converting the universal proposition, and that A belongs to B, which was the conclusion, it has indeed been shown that A belongs to some C; but that it belongs to some B has not been shown, although it is necessary that if A belongs to some B, it also belongs to some C. But it is not the same thing for this to belong to that and for that to belong to this; rather, it must be assumed that if this belongs to some of that, the other also belongs to some of this. But once this is assumed, there is no longer a syllogism from the conclusion and the other proposition. But if B belongs to every C and A to some C, it will be possible to show that A belongs to C when it is assumed that B belongs to every C and A to some C; for in this way, if B belongs to every C and A to some B, it is necessary that A belong to some C, with B as the middle term. And when one of the propositions is affirmative and the other privative, and the universal one is affirmative, the other will be shown. For let B belong to every C and let A not belong to some C; the conclusion is that A does not belong to some C. If, then, it is assumed that B belongs to every C, and A belongs to some B, it is necessary that A not belong to some C, with B as the middle term. But when the universal proposition is privative, the other is not shown except as in the earlier cases, if it is assumed that what this does not belong to some of, the other belongs to some of, as if A belongs to no C and B to some C; the conclusion is that A does not belong to some C. If, then, it is assumed that what B does not belong to some of, A belongs to the same some of, it is necessary that A belong to some C; but otherwise, by converting the universal proposition, it is not possible to show the other, for in no way will there be a syllogism.
Manifestum igitur quoniam in prima quidem figura per se invicem est ostensio, et per primam,et per tertiam figuram fit: nam cum prædicativa quidem estconcIusio, perprimam,cum autem privativa, per postremam, sumitur enim cui hoc nulli, aHerum ornni,inesse. In media autem, cum universalis est quidem syllogismus et per ipsarn,et perprimam figuram, etper postremam; cum autem particularis, et per ipsam, et per postremam. In tertia vero per ipsam,;:omnes. Manifestum etiam quoniam in media et in tertia qui non per ipsas fiunt sylIogisrnus,aut non sunt secundum eam qure circulo est ostensionem, aut imperfecti sunt.
CAPUT VIII.
De syllogismo conversiO.
Convertere autem est transponentem conclusionem facere syllogismum, quoniam vel extremum medio non inerit, vel hoc postremo;necesse est enim conc 'usione conversa,et altera remanente propositione, interimi reliquam; nam si erit, et conclusio erit: ditfert autem opposite aut contrarie convertere conclusionem, non enim fit idem syllogismus utrolibet conversa; paIam autem bocerit persequentia.Dico autem opponi quidem omni inesse non omni, et alicui nulli,contrarie autem omni nuIli,et alicui non alicui inesse.Sit enim ostensum a de per me dium; si igitur sumatur a nulli inesse,omni autem b,nuIli inerit b,et si a quidem nulli e, autem omni c,a non omni b,et non omnino nulIi,non enim ostendebatur universale per tertiam figuram.
It is clear, then, that in the first figure the proof proceeds through the propositions themselves, and it comes about both through the first and through the third figure; for when the conclusion is affirmative, it is through the first, but when it is privative, through the last—for one takes that which belongs to no one of this, and the other that it belongs to every one. In the middle figure, however, when the syllogism is universal, it comes about both through itself and through the first figure and through the last; but when it is particular, both through itself and through the last. In the third figure, however, all come about through itself. It is also clear that in the middle and in the third figure the syllogisms that are not produced through the figures themselves are either not according to that demonstration which is by circular proof, or are imperfect.
CHAPTER VIII
On the conversion of the syllogism.
To convert is to make a syllogism by transposing the conclusion, showing that either the extreme does not belong to the middle, or this does not belong to the last; for with the conclusion converted and one of the premises remaining, the other must be destroyed; for if it remains, the conclusion will also remain. But it makes a difference whether one converts the conclusion oppositely or contrarily, for the same syllogism does not result whichever way it is converted; that this will be clear from what follows. I say that "to belong to every" is opposed to "not to belong to every," and "to belong to some" to "to belong to none"; but contrarily, "to belong to every" to "to belong to none," and "to belong to some" to "not to belong to some." For let it have been shown that A belongs to no B through the middle; if, then, it is assumed that A belongs to no B, and that it belongs to every C, then B will belong to no C; and if A belongs to no C, but to every B, then A will not belong to every C, and not entirely to none; for the universal was not shown through the third figure.
Omnino autem eam quæ est ad majorem extremitatem propositionem non est detruere universaliter per conversionem,semper enim interimitur per tertiam figuram,necesse enim ad postremam extremitatem utrasque sumere propositiones.Et si privativus sit syllogismus similiter: ostendatur enim a nulli inesse per b,ergo si sumatur a omni inesse, nulli autem b,nuIli inerit b. Et si a et omni c,a alicui b,sed nulli inerat.Si autem opposite convertatur conclusio, et alii syllogismi oppositi, et non universales erunt,fit enim altera propositio particularis, quare conclusio erit particularis. Sit enim praedicativus syllogismus, et convertatur sic, ergo si a non omni c,b autem cmni b,non omni c. Et si a quidem non omni c,b autem omni a,non omni b. Similiter autem et si privativus sitsyllogismus,nam si a alicui inest,b autem nulIi,b alicui non inerit, et non simpliciter nulIi, et si a quidem alicui c, autem omni,quemadmodurn in principio sumptum est,a alicui inerit.ln particularibus autem syJlogismis quando opposite convert itur concJusio,interimuntur utræque propositiones,quando vero contrariæ,neutra; non enim jam accidit quemadmodum in universalibus interimere deficiente conclusione secundum conversionem,sed nec omnino interimere. Ostendatur enim a de aliquo per; ergo si suma.
But in general the proposition that relates to the major extreme cannot be destroyed universally by conversion; for it is always refuted through the third figure, since it is necessary to take both propositions in relation to the last extreme. And similarly if the syllogism is privative; for let it be shown that A belongs to no B; therefore if it is assumed that A belongs to every B, and to no B, then B will belong to no B. And if A belongs to every C, and A to some B, but belonged to no B. But if the conclusion is converted to the opposite, the other opposite syllogisms also result, and they will not be universal; for one of the propositions becomes particular, and therefore the conclusion will be particular. For let the syllogism be affirmative, and let it be converted thus: therefore if A does not belong to every C, but B belongs to every C, then A does not belong to every C. And if A does not belong to every C, but B belongs to every A, then A does not belong to every B. Similarly also if the syllogism is privative; for if A belongs to some B, but to no B, then B will not belong to some B, and not simply to none. And if A belongs to some C, but to every B, as was assumed at the beginning, A will belong to some B. But in particular syllogisms, when the conclusion is converted to the opposite, both propositions are destroyed; but when to the contrary, neither; for it no longer happens, as in universal syllogisms, to destroy by conversion when the conclusion is lacking, but not to destroy at all. For let it be shown that A belongs to some B; therefore if it is assumed...
tur a nulli inesse,b autem alicui c,a alicui non inerit,et si a nulli c,b autem omni, nulli inerit b; quareinterimentur utræque.Si autem contrarie convertantur,neutra; nam si a alicui non inest,b autem omni,b alicui non inerit, sed nondum interimitur quod ex principio, contingit enim alicui inesse,et alicui non inesse: universali autem sublato a b,omnino non fit syllogismus. Si enim a quidem alicui non inest,b autem alicui inest, neutra propositionum universalis est. Similiter autem et si privativus sit syllogismus,si enim sumatur a omni inesse,interimuntur utræque; si autem alicui,neutra; demonstratio autem eadem.
CAPUT IX.
De syllogismo conversivo ill secunda figura,
In secunda autem figura,earn quidem quae est ad majorem extremitatem propositionem, non est interim ere contrarie,quolibet modo conversione facta, semper erit conclusio in tertia figura, universalis autem non fuit in hac syllogismus, alteram autem in hac interimemus,similiter conversione.Dico autem similiter: si contrarie quidem convertitur, contrarie; si opposite,opposile. Insit enim a omni b, autem nulli,conclusio c.Si erga sumatur omni inesse et a maneat,a omni inerit,fitenim prima figura.
But if it is converted with respect to neither, and B belongs to some C, then A will not belong to some C; and if A belongs to no C, but B to all, then B will belong to none; hence both propositions will be destroyed. But if they are converted contrarily, neither will be destroyed; for if A does not belong to some, but B to all, then B will not belong to some, but the original conclusion is not yet destroyed, for it is possible for it to belong to some and not to some; but when the universal proposition is removed, no syllogism at all comes about. For if A does not belong to some and B belongs to some, neither proposition is universal. Similarly also if the syllogism is privative: for if A is taken to belong to all, both propositions are destroyed; but if to some, neither; and the demonstration is the same.
CHAPTER IX.
On the convertible syllogism in the second figure.
In the second figure, the proposition that relates to the major extreme cannot be destroyed by a contrary conversion in any way, since the conclusion will always be in the third figure, but there was no universal syllogism in this figure; but we shall destroy the other proposition in this figure, similarly by conversion. I mean similarly: if it is converted contrarily, contrarily; if oppositely, oppositely. For let A belong to all B, or to none, and let the conclusion be C. If then it is taken to belong to all and A remains, A will belong to all, for the first figure comes about.
Si autem omni c, a autem nulli c, a non omni b, figura postrema.Si autem opposite convertatur c,a quidem similiter ostendetur,a autem opposite: nam si alicui c,a autem nulli c,a alicui non inerit; rursum si alicui c,a autem omni b,a alicui c, quare oppositus fit syllogismus. Similiter autem ostendetur et si e converso se habeant propositiones.Si autem particularis est syllogismus, contrarie quidem conversa conclusione neutrapropositionurn interimitur,quemadmodum nec in prima flgura,opposite autem, utræque. Ponatur enim a quidem nulli inesse, autem alicui, conclusio c. Si igitur ponatur alicui inesse, et a maneat conclusio erit quoniam a alicui non inesl,sed non interimitur quod ex principio, contingit enim alicui inesse et non inesse.Rursum si alicui c,et a alicui c,non erit syllogismus, neutrum enim universale eorum quae sumpta sunt,quare non interimitur a b.Si autem opposite convertatur,interimuntur utræque, non si omni c,a autem nulli b,nulli c,a erit autem alicui. Rursum si omni c,aautemalicuic,alicui b,a*.Eadem autem demonstratio et si universalis sit prædictiva.
But if A belongs to all C and to no C, then A does not belong to all B: the last figure. But if the opposite is converted, C, A will indeed be shown similarly, but the opposite: for if A belongs to some C and to no C, then A will not belong to some; again, if A belongs to some C and to all B, then A belongs to some C, so the syllogism becomes opposite. It will be shown similarly also if the propositions are related conversely. But if the syllogism is particular, when the conclusion is converted contrarily, neither proposition is destroyed, just as neither was in the first figure; but when converted oppositely, both are destroyed. For let it be posited that A belongs to none, but to some, and the conclusion is C. If, then, it is posited that it belongs to some, and A remains, the conclusion will be that A does not belong to some, but the original assumption is not destroyed; for it is possible for it to belong to some and not to belong. Again, if A belongs to some C and to some C, there will be no syllogism; for neither of the things assumed is universal, so A B is not destroyed. But if the opposite is converted, both are destroyed: if A belongs to all C and to no B, it will belong to no C, but it will belong to some. Again, if A belongs to all C and to some C, then A belongs to some B. The same demonstration holds also if the predicative proposition is universal.
CAPUT X.
De syllogismo conversivo in tertia figura,
In tertia vero figura quandocontrariequidem convertitur conclusio,neutra propositionem interimitur secundum nullum syllogismorum; quando autem op. posite,utræque in omnibus.Si enim ostensum a alicui inesse,medium autem sumptum c, et sint llniversales propositi ones, si ergo sumatur a alicuib non inesse,b autem omni c,non fit syllogismus ejus quod est a de c.Neque si a alicui non inest,c autem omni,non erit ejus quod est syIlogismus.SimiIiter autem ostendetur et si non universales sint ropositiones,aut enim utrasque necesse est particulares esse per conversionem,aut universalem ad minorem extremitatem fieri, sic autem non fiet syIIogismus,nec in prima figura,nec in media.Si autem opposite convertantur propositiones, interimuntur utræque, nam si a nulli b, autem omni c, a nulli c.Rursum si a quidem nulli,c autem omni,b nulli c.Et si altera non sit universalis, similiter; si enim a nulli b,b autem alicui c,a alicui non inerit.Si autem a quidem nulli, autam omni, nulli c, b. Similiter etsi privativus sit syllogismus; ostendatur enim ctalicui non inesse;si autem praedicativa quidem c, cac autem negativa, sic enim fiebat syIlogismus. Quando igiturcontrarium sumitur conclusioni non eritsyllogismus, nam si a alicui b, autem omni c, non fit syllogismus ejus quod est a et c.
CHAPTER X.
On the convertible syllogism in the third figure.
In the third figure, when the conclusion is converted into its contrary, neither premise is destroyed by any of the syllogisms; but when it is converted into its contradictory, both premises are destroyed in all cases. For if A has been shown to belong to some B, and the middle term taken is C, and the premises are universal, then if it is assumed that A does not belong to some B, but B belongs to all C, no syllogism arises concerning A and C. Nor, if A does not belong to some C, but C belongs to all, will there be a syllogism concerning them. The same will be shown even if the premises are not universal; for either both must be particular by conversion, or the universal must be made the minor extreme, and in this way no syllogism will arise, neither in the first figure nor in the middle. But if the premises are converted into their opposites, both are destroyed: for if A belongs to no B, but B to all C, then A belongs to no C. Again, if A belongs to no C, but C to all, then B belongs to no C. And if one premise is not universal, it is likewise; for if A belongs to no B, but B to some C, then A will not belong to some C. But if A belongs to no B, but B to all C, then A belongs to no C. So too if the syllogism is privative; for it is shown that C does not belong to some. But if the premise concerning C is affirmative and that concerning A negative, for thus the syllogism came about. When, therefore, the contrary of the conclusion is assumed, there will be no syllogism; for if A belongs to some B, but B to all C, no syllogism arises concerning A and C.
Neque si a alicui b, nulli autem c, non fuit ejus quod nst a etcsyllogismus, quare non interimuntur propositiones. Quando vero oppositum, interimuntur; nam si a omni, 6, etomni c, a omni c, sed nulli inerat. Rursum si a cmni b, nulli autem c, nulli c, sed omni inerat. Similiter aulem monstratur, et si non universales sint propositiones: sit enim a universalis et privativa, altera autem parti;,ulariset in prædicutiva ergo si a quidem omni b, autem alicui c, a alicui accidit, sed nulli inerat. Rursum si a omni nulli autem et nulli c. Si autem a alicui b, et alicui c, non fit syllogismus. Neque si a alicuib. et nulli c, nec sic. Quare illo quidem modo interimuntur, sic autem non interimunturpropositiones. Manifestum est ergo ex iis quae dicta sunt quomodo conversa eonclusione in unaquaque figura fit syllogismus, et quando contrarie propositioni, et quando opposite: et quoniam in prima quidem figura per mediam et postremam fiunt syllogismi, etquæ quidem ad minorem extremitatem semper per mediam interimitur, quae veru ad majorem per postremam; in secunda autem, per primam et postremam, quae quidem ad minorem extremitatem semper per prirnam figuram quæ vero ad majorem, per postremam; in tertia vero, per primam et per mediam, et quæ quidem ad majorem per primam semper, quæ vero ad minorem per mediam semper.
Nor, if a belongs to some b but to no c, was there a syllogism of the proposition that a belongs to no c; therefore the premises are not destroyed. But when the opposite is assumed, they are destroyed; for if a belongs to all b and to all c, then a belongs to all c, but it belonged to none. Again, if a belongs to all b but to no c, then it belongs to no c, but it belonged to all. The same is shown also if the premises are not universal: for let a be universal and privative, and the other particular and predicative; then if a belongs to all b but to some c, a will belong to some c, but it belonged to none. Again, if a belongs to all b and to no c, then it belongs to no c. But if a belongs to some b and to some c, no syllogism results. Nor if a belongs to some b and to no c, neither thus. Therefore in that way the premises are destroyed, but in this way they are not destroyed. It is clear, then, from what has been said, how, when the conclusion is converted, a syllogism is produced in each figure, and when it is produced against the contrary proposition and when against the contradictory; and that in the first figure syllogisms are produced through the middle and the last figure, and those directed at the minor extreme are always destroyed through the middle figure, while those directed at the major extreme are destroyed through the last; in the second figure, through the first and the last, those directed at the minor extreme always through the first figure, but those at the major through the last; in the third figure, through the first and the middle, those directed at the major extreme always through the first, but those at the minor always through the middle.
Quid ergo est convertere, et quomodo in unaquaque figura, et quis fit syllogismus manifestum.
CAPUT XI.
De syllogismo per impossibile.
Perimpossibile autem syllogismus ostendil ur qui. dem, quando contradictio ponitur conclusionis, et assumitur altera propositio. Fit autem in omnibus figuris, simile enim est conversioni. Verumtamen differt in tantum quoniam convertitur quidem facto syllogismo,et sumptis utriusque propositionibus. Deducitur autem ad impossibile non confesso opposito prius, sed manifesto quoniam est verum. Termini vero similiter se habent in utriusque, eteademsumptio utrorumque, ut si a inest omni b, medium autem c, si supponitur a non omni vel nulli inesse, vero, omni, quod fuit verum, necese est aut nulli aut nonomniinesse, hoc autem impossibile, quare falsum estquodsuppositurnest. Verumergo oppositum; si. militer autem in aliis figuris, quaecunque enim conversionem suscipiunt, et per impossibile syIlogismum. Ergo aliaquidempropositaomnia ostenduntur per impossibilein omnibus Hguris, universale autem praedicativum in media et in tertia monstratur, in prima autem non monstratur: supponatur enim a non omni baut nulliinesse,et assumatur alia propositio, utrolibetmodo, sive a omni inest c, sive omni (SIC enim erat prima figura); si ergo supponatur a non omni inesse, non fiet syllogismus quomodolibet sumpta propositiones.
What conversion is, then, and how it works in each figure, and what syllogism results, is clear.
CHAPTER XI.
On syllogism through the impossible.
A syllogism through the impossible proves its point when the contradictory of the conclusion is posited and the other premise is assumed. It occurs in all the figures, for it is similar to conversion. Yet it differs to this extent, that conversion is effected after the syllogism has been made and both premises have been taken; but the reduction to the impossible is not made when the opposite has first been conceded, but when it is evident that it is true. The terms stand in the same relation in both, and the assumption of both is the same; for example, if a belongs to every b, and c is the middle term, if it is supposed that a belongs not to every b or to no b—which was true, namely that it belongs to every b—it is necessary that a belong to no c or not to every c; but this is impossible, wherefore what was supposed is false. The opposite, then, is true; and similarly in the other figures, for whatever admit of conversion also admit of syllogism through the impossible. Therefore all the other propositions are proved through the impossible in all the figures, but the universal affirmative is proved in the middle and in the third figure, while in the first it is not proved; for let it be supposed that a belongs not to every b or to no b, and let the other premise be assumed in either way, whether that a belongs to every c or to no c (for such was the first figure); if, then, it is supposed that a belongs not to every b, there will be no syllogism, however the premises are taken.
Si autem nulli bd, quidem assumatur, syllogismus quidem erit falsi, non ostenditur autem propositum; nam si a nulli b, autem omni d, a nulli d, buc autem sit impossibile, falsum igitur est nulli inesse a, sed non si nulli falsum, omni verum. Si autem ca assumatur, non fit syltogismus, nec quando supponiturnon omni inesse n; quare manifestum quoniam omni inesse non ostendttu!" in prima figura per impossibiIe.AIicui autem, et nulli, et nun omni osienditur. Supponatur enim a nulli inesse, autem sumptum si omni aut alicui ergo necesse est a nulli aut non omni inesse, hoc autem impossibile. Sit enim verum et manifestum quoniam omni inest a, quaresi hoc falsum, necesse est a alicui inesse. Si autem ad a sumatur altera propositio, non erit syllogismus, neque quando sub contrarium conclusioni supponitur ut alicui non ines se; manifestum ergo quoniam oppositum sumendum est. Rursum supponatur a alicui inesse, sumptum autem eit omni a, necesse est igitur alicui inesse, boc autem sit impossibile, quare falsum quidem suppositum est;si autem sic, verum est nulli inesse Similiter autem et si privativa sumpta sit ca.Si autem ad sumptasit propositio,non erit syllogismus.
But if it is assumed that A belongs to no B, and to all D, then there will indeed be a syllogism of a falsehood, but the proposed conclusion is not proved; for if A belongs to no B and to all D, then A belongs to no D. But let it be impossible that B belongs to no D; therefore it is false that A belongs to no B. Yet it is not the case that if this is false, then it is true that it belongs to all; but if the assumption concerning C is made, no syllogism results, nor when it is supposed that it does not belong to every N. Hence it is clear that belonging to every one is not proved in the first figure through the impossible. But belonging to some and to none and not to every one is proved. For let it be supposed that A belongs to no B, and let the other premise be assumed as belonging to every one or to some one; then it is necessary that A belong to no one or not to every one, but this is impossible. For let it be true and evident that A belongs to every one; therefore, if this is false, it is necessary that A belong to some one. But if the other premise is taken in relation to A, there will be no syllogism, nor when the contrary of the conclusion is supposed, namely that it belongs to some one not. It is clear, therefore, that the opposite must be assumed. Again, let it be supposed that A belongs to some one, and let the premise assumed be that it belongs to every one; then it is necessary that it belong to some one, but let this be impossible; therefore the supposition is indeed false; but if so, it is true that it belongs to no one. Similarly also if the privative premise is assumed. But if the premise is taken in relation to the assumption, there will be no syllogism.
Si autem contrarium supponatur, syIlogismus eritet impossibile, non tamen ostenditur quod est propositum: supponatur enim o omni 6, et sumptum sit omni a, ergo necesse estc omDi inesse: hoc autem impossibile, quare falsum est omni inesse a, sed nondum erit necessarium, si non omni, nuUnnesse. Similiter autem et si a sumaturaltera proposito: nam syllogismus quidem eritet irnpossibile,non interimitur autem hypothesis, quare oppositum supponendurn. Ad ostendendum autem non omni binesse a; supponendum omni inesse, nam si a omni b, etc omni a, omni inerit: si ergo hoc impossibile, falsum quod suppositurn est;; similiter autem etsi ad sumpta sit altera propositio. Et si privativa sit a, similiter, nam et sic fit syllogismus. Si autem ad sumpta sit privativa, nihil ostenrlitur, Si autem non omni, sed alicui inesse supponatur, non ostenditur quoniam non omni, sed quoniam nulli: si enim a alicui b, autem omni a, alicui inerit c; si ergo hoc impossibile, falsum est alicui inesse a, quare verum nulIi; hoc autem ostenso, interirniturverurn, nam a alicui quidem inerat,alicui vero non inerat, Amplius autem non tam propter hypothesin accidit impossibile, falsaenim erit, siquidem ex veris non est falsum syllogizare: nunc autemest vera,inestenim a alicui b, quare non supponendum alicui inesse, sed omni.
But if the contrary is supposed, there will be a syllogism of an impossibility, yet what was proposed is not proved. For let it be supposed that A belongs to all B, and let it be assumed that A belongs to all C; then it is necessary that B belong to all C. But this is impossible; therefore it is false that A belongs to all B. But it will not yet be necessary that, if A does not belong to all B, it belongs to none. Similarly also if the other proposition is assumed. For there will indeed be a syllogism of an impossibility, but the hypothesis is not refuted; therefore the opposite must be supposed. But to show that A does not belong to all B, one must suppose that it belongs to all B; for if A belongs to all B and to all C, then B will belong to all C. If therefore this is impossible, what was supposed is false. Similarly also if the other proposition is assumed. And if A is privative, similarly; for in this way too a syllogism is produced. But if the assumed proposition is privative, nothing is shown. But if it is supposed that A does not belong to all B but to some, it is not shown that it does not belong to all, but that it belongs to none. For if A belongs to some B, and to all C, then B will belong to some C. If therefore this is impossible, it is false that A belongs to some B; therefore it is true that it belongs to none. But when this has been shown, the truth is refuted; for A belonged to some and did not belong to some. Moreover, the impossibility does not arise so much on account of the hypothesis; for it will be false, since it is not possible to syllogize a falsehood from true premises. But now it is true; for A belongs to some B. Therefore one must not suppose that it belongs to some, but to all.
Similiter autem et si alicui non inest a, ostenderemus: si enim idem est alicui non in esse, et non omni inesse, eadem in utrisque demonstratio. Manifestum ergo quoniam non contrarium, sed oppositum supponendum in omnibus syllogismis, sic enim necessarium erit et axioma probabile; nam si de omni vel affirmatio vel negatio, ostenso quoniam non negatio, necesse est affirmationem veram esse; rursum si non ponant veram esse affirmationarn, constat veram esse negationem; contrariam vero neutro modo contingit ratum facere. Neque enim Į necessarium, si nulli falsum, omni verum, neque probabile ut sit alterum falsum, quoniam alterum varum. Manifestum ergo quoniam in prima figura alia quidem proposita omnia ostenduntur per i1.1possibile, universale autem affirmativum non ostenditur.
CAPUT XII.
De syllogismo per impossibile in secunda figura,
Similarly, also if we show that A does not belong to some one, since to belong to some one not and not to belong to every one are the same, the same demonstration will apply in both cases. It is clear, therefore, that in all syllogisms one must suppose not the contrary but the opposite; for thus it will be both necessary and a probable axiom. For if of every thing there is either affirmation or negation, then when it has been shown that the negation does not hold, the affirmation must be true; again, if they do not grant that the affirmation is true, it is evident that the negation is true. But it is in no way possible to establish the contrary. For it is not necessary that, if it is false of none, it is true of all; nor is it probable that one is false because the other is true. It is clear, therefore, that in the first figure all other propositions are proved through the impossible, but the universal affirmative is not proved.
CHAPTER XII.
On the syllogism through the impossible in the second figure,
In media autem figura et postrema et hoc ostenditur. Ponaturenim a non oami inesse, sumptum sit autem omni inesse a; ergo si quidem non omni inest a, autem omni, non omni inest c, hoc autem impossibile. Sit enim manifestum quoniam omnib inest c, quare falsum quod suppositum est, verum est ergo omni inesse. Si autem contrarium supponatur, syllogismus quidem erit ad impossibile, non tamen ostenditur quod propositum est. Si enim a nulli b, omni autem c, nulli 6, c, hoc autem impossibile, quare falsum est, nulli inesse, sed non si hoc falsum, verum omni. Quando autem alicui inest a, supponatur a nulli inesse, autem omni insit; necesse est ergoc nulli inesse, quare si hoc impossibile, necesse est a alicui inesse. Si autem supponatur alicui non esse, eadem erunt quae in prima figura, Rursum supponatur a alicui inesse, autem nulliinsit necesse est igiturc alicui non inesse v sed omniinerat, quare falsum quod suppositum est, nulli ergo inerat a. Quando autem non omni inest a, supponatur omni inesse: autem nulli, necesse est ergo nulli inesse, hoc autem impossibile, quare verum est non omni inesse. Manifestum ergo quoniam omnes syllogismi fiunt per mediam figuram.
But in the middle and last figure this also is shown. For let it be supposed that A does not belong to every B, and let it be assumed that A belongs to every C; therefore, if A does not belong to every B, but belongs to every C, then C does not belong to every B; but this is impossible. For let it be manifest that C belongs to every B; therefore the supposition was false, and so it is true that A belongs to every B. But if the contrary is supposed, there will indeed be a syllogism to an impossibility, yet what was proposed is not shown. For if A belongs to no B, but to every C, then C belongs to no B; but this is impossible; therefore it is false that A belongs to no B. But it does not follow that, if this is false, it is true that A belongs to every B. But when A belongs to some B, let it be supposed that A belongs to no B, and let it be assumed that A belongs to every C; then it is necessary that C belongs to no B; therefore, if this is impossible, it is necessary that A belongs to some B. But if it is supposed that A does not belong to some B, the same results will follow as in the first figure. Again, let it be supposed that A belongs to some B, and let it be assumed that A belongs to no C; it is necessary, therefore, that C does not belong to some B; but it belonged to every B; therefore the supposition was false; therefore A belonged to no B. But when A does not belong to every B, let it be supposed that A belongs to every B, and let it be assumed that A belongs to no C; it is necessary, therefore, that C belongs to no B; but this is impossible; therefore it is true that A does not belong to every B. It is manifest, therefore, that all syllogisms are produced through the middle figure.
CAPUT XIiI.
De ostensione per impossibile in tertia figura,
Similiter autem et per ultimam. Ponatur enim a alicui non inesse, autem omni b, ergo a alicui non iaerit; si ergo boc impossibile, falsum aliLmi non nesse, quare verum est omni. Si vero supponatur nulli inesse, syllogismus quidern edt, etimpossibiIe, non ostendit autem quod propositum est; si enim contrarium supponatur, eadem eruntquæ in prioribus. Sed ad ostendendum alicui inesse, eadem sumenda est hypothesis, nam si a nulli b, autem alicui b, a non omni c: si ergo hoc falsum, verum esta alicui inesse. Quando autem nulli inest a, supponatur alicui inesse, sumptum sit autem et omni inesse ergo, necesse est a alicui inesse; sed nulli inerat, quare falsum est alicui inesse a. Si autem supponatur omni inesse a, non ostenditur propoaitum: sed ad ostenden dum non omni inesse, eadem sumenda hypothesis, nam si a omni 5, et alicui b, a inest alicui; hoc autem non fuit, quare faI. sum est omni in esse, si autem sic, verum non omni. Si autem supponatur alicui inesse, eadem erunt quæ et in iis quæ prius dicta sunt. Manifestum ergo quoniam in omnibus per impossibile syllogismis oppositum supponendum. Palam,autern et quoniam in media figura ostenditur quodammodo affirmativurn, et in postrema universale.
CHAPTER XIII.
On proof through impossibility in the third figure.
Likewise also through the last figure. For let it be supposed that A does not belong to some [B], but [that A belongs] to every B; therefore A will not belong to some [B]. If, then, this is impossible, it is false that it does not belong to some; hence it is true that it belongs to every. But if it is supposed that it belongs to none, there is indeed a syllogism and an impossibility, but it does not show what was proposed; for if the contrary is supposed, the same things will result as in the earlier cases. But to show that it belongs to some, the same hypothesis must be assumed; for if A belongs to no B, but [A belongs] to some B, [then] A does not belong to every C. If, then, this is false, it is true that it belongs to some. But when A belongs to none, let it be supposed that it belongs to some; and let it also have been assumed that it belongs to every; therefore it is necessary that A belong to some; but it belonged to none, so it is false that A belongs to some. But if it is supposed that A belongs to every, what was proposed is not shown; but to show that it does not belong to every, the same hypothesis must be assumed; for if A [belongs] to every [B], and [A belongs] to some B, A belongs to some [C]; but this was not so, so it is false that it belongs to every; but if so, it is true that it does not belong to every. But if it is supposed that it belongs to some, the same things will result as in those which were said before. It is clear, therefore, that in all syllogisms through impossibility the opposite must be supposed. It is evident, moreover, also that in the middle figure the affirmative is in a certain way shown, and in the last figure the universal.
CAPUT XIV.
Quo justa, et quæ ad impossibile ducit demonstratio, differant.
Differt autem quae ad impossibile demonstratio ab ea quae est oSlensiva,eo quod ponat quod vult intenmere, deducens ad confessum falsum, ostensiva autem incipit aconfessis positionibus veris. Sumunt ergo utræque duas propositiones confessas,sed haec quidem ex quibus est syllogismu5, illa vero unam quidem baru m, alteram vero contradictionem concIusionis. Et hinc quidem non necesse est notam esse conclusionem,neque prius op inari quoniam est, aut non est; illinc vero necesse est, quoniam non est. Differt autem nihil affirmativam, vel negativam esse concIusionem. sed similiter se habet in utrisque. Omnis enim quæ ostensive concluditur, et per impossibile monstrabitur, et quae per impossibile ostensive, et per eosdem terminos, non autem in eisdem figuris. Nam quando per impossibile syllogismus fit in prima figura, quod verum est in media erit, aut in postrerna, privativum quidem in media, prædicativum autem in postrema. Quando autem syIlogismus in media fit, quod verum est erit in prima figurain omnibus propositionibus, quando autem in postrerna syllogism us, quod verum est erit in prima et in media, affirmativa quidem in prima, privativa autem in media.
CHAPTER XIV.
How the direct demonstration and the one leading to impossibility differ.
The demonstration leading to impossibility differs from the ostensive one in that it posits what it wishes to establish, deducing to an admitted falsehood, whereas the ostensive begins from admitted true positions. Both, then, take two admitted propositions, but the latter takes those from which the syllogism is formed, while the former takes one of these and the other the contradictory of the conclusion. And in the latter case it is not necessary that the conclusion be known beforehand, nor that it be previously opined that it is or is not; but in the former it is necessary, since it is not. It makes no difference whether the conclusion is affirmative or negative, but matters stand alike in both. For every conclusion reached ostensively will also be shown through impossibility, and every one shown through impossibility ostensively, and by the same terms, though not in the same figures. For when the syllogism through impossibility is formed in the first figure, what is true will be in the middle or in the last figure—privative in the middle, but predicative in the last. But when the syllogism is formed in the middle figure, what is true will be in the first figure in all the propositions; but when the syllogism is in the last figure, what is true will be in the first and in the middle—affirmative in the first, but privative in the middle.
Sit enim ostensum a nulli aut non omni per primam figuram, ergo hypothesis quidem erat alicui inesse a, autem sumebatur a quidem omni inesse, autem nulli sic enim fiebat syUog!smus ad impossibile. Hoc autem media figura, si a quidemomni,b!lutem nulli, inest, etmanifestumex his quoniam nulli inest a. Similiter autem et si non omni ostensum sit inesse, nam bypothesis quidem est omni a messe, autam sumebatur a quidem omni, autem non omni, et si privativa sit sumpta a, similiteretenimsic fitin media figura. Rursum sit ostensum alicui inesse a, ergo hypothesis quidem est nulli inesse, autem sumebatur omni in esse, et a vel omni vel alicui c, sic enim erit impossibiIc. Hoc autem postrema figura, si a et omni c, et manifestum ex his quia necesse est a alicui binesse, similiter autemet si alicui csumatur inesse vel a. Rursum in media figura ostensum sit a omni inesse, ergo hypothesis quidem fuit, non omni inesse a, sumptum est autem a omni C, et omni b, sic enim erit impossibile; boc autem prima figura, si a omni c, et omni b. Similiter autem et si ostensum sit alicui inesse, nam hapotbesis quidem fuit, nulli inesse a, sumptum est autem a omni c, et alicui b.
For let it have been shown by the first figure that A belongs to none or not to all; therefore the hypothesis was that A belongs to some, but it was assumed that A belongs to all, and [that it belongs] to none—for thus the syllogism to the impossible came about. But this [happens] in the middle figure, if A belongs to all and B to none, and it is evident from these that A belongs to none. Similarly also if it has been shown that it does not belong to all; for the hypothesis was that A belongs to all, but it was assumed that A belongs to all and [that it belongs] not to all; and if the privative premise is assumed, it is similarly so in the middle figure. Again, let it have been shown that A belongs to some; therefore the hypothesis was that it belongs to none, but it was assumed that it belongs to all, and A [belongs] either to all or to some C—for thus it will be impossible. But this [happens] in the last figure, if A [belongs] to all C, and it is evident from these that A must belong to some B; similarly also if it is assumed that it belongs to some C or to A. Again, in the middle figure let it have been shown that A belongs to all; therefore the hypothesis was that A does not belong to all, but it was assumed that A belongs to all C and to all B—for thus it will be impossible; but this [happens] in the first figure, if A [belongs] to all C and to all B. Similarly also if it has been shown that it belongs to some; for the hypothesis was that A belongs to none, but it was assumed that A belongs to all C and to some B.
Si autem privativus fit syllogismus, hypothesis quidem a alicui inesse, sumptum est autem a nulli c, et omni b, quare fit prima figura, Et si non universalis sit syllogismus, sed a alicui ostensum sit nrn inesse, similiter:nam hypothesis quidem omnib inesse a, sumptum estautem a nulli c, et alicui b, sic enim prima figura, Rursum in tertia figura ostensum sit a inesse omni b, ergo hypotbesis quidem fuitnon omni inesse a, sumptum est autem omni,b, et a omni c, sic enim erit impossibila,boc autem prima figura. Similiter autem et si in aliquo sit demonstratio,non hypothesis qui. dem erit nulli,b inesse a,sumptumest autem alicui et a omni c,Si autem privativus sit syllogismus, hypothesis quidem a alicui inesse, sumptum est autem ftquidem nulli, autem omni, boc autem media figura. Similiter autem etsi non universalis sit demonstratio, nam hypothesisquidem eritomni inesse a, sumptum est autem a quidem nulli, autem alicui, hoc autem media figura. Manif'estum ergo quoniam per eosdem terminos et ostensive est demonstrare unumquodque propositum, et per impossibile. Similiter autem erit, et cum sint ostensivi s;yllogismi,ad impossibile deducere in terminis sumptis, quando opposita propositio conclusioni sumpta fuerit, nam fiunt iidem syllogismi.
But if the syllogism becomes privative, the hypothesis was that A belongs to some, but it was assumed that A belongs to no C and to all B; for thus the first figure is produced. And if the syllogism is not universal, but it has been shown that A belongs to some, similarly: for the hypothesis was that A belongs to all B, but it was assumed that A belongs to no C and to some B; for thus the first figure. Again, in the third figure let it have been shown that A belongs to all B; therefore the hypothesis was that A does not belong to all, but it was assumed that [B belongs] to all and A to all C; for thus it will be impossible, and this is the first figure. Similarly, moreover, if the demonstration is that it belongs to some, the hypothesis will be that A belongs to none B, but it was assumed that [B belongs] to some and A to all C. But if the syllogism is privative, the hypothesis was that A belongs to some, but it was assumed that [B] belongs to none, or [A] to all; and this is the middle figure. Similarly, moreover, even if the demonstration is not universal: for the hypothesis will be that A belongs to all, but it was assumed that A belongs to none, or [B] to some; and this is the middle figure. It is evident, therefore, that through the same terms it is possible to demonstrate each proposed conclusion both ostensively and through impossibility. Similarly, moreover, it will also be the case, when the syllogisms are ostensive, to reduce to impossibility with the terms assumed, when the proposition opposite to the conclusion has been assumed; for the same syllogisms are produced.
iis qui sunt per convesionem, quare statim habemus et figuras per quas unumquodque erit.Palarn ergo quoniam omne propositum ostenditur per utrosque modos et per impossiblle et ostensive, et non contingit separari alterum ab altero.
CAPUT XV.
De ratiocinatione ex oppositis.
In qua autem figura est ex oppositis propositionihus syllogizare, et in qua non est,sic erit manifeslum.Dico autem oppositas esse propositiones, secundum locut.ionem quidem quatuor, ut omni et,n ulli et omni et non omni, et aliqui et nulli, et alicui et non alicui inesse; secundum veritatem autem Ires, nam alicui et non alicui secundum locutionem opponuntur solum; harum autem contrarias quidem universales, omni nulli inesse, ut omnem disciplinam esse studiosam, nullam esse studiosam, alias vero oppositas.ln prima igitur figura non est ex oppositis propositionibus syllogismus, neque affirmativus, neque negativus; affirmativus quidem,quoniam oporlel utrasque affirmativas esse propositiones,oppositæ autem affirmatio et negatio; privativus autem,quoniarn oppositas quidem idem de eodem praedicant et negant,in trimaautem medium non dicitur de utrisque, sed de illo quidem aliud negatur,idem autem de alio prædicatur, hae vero non opponuntur.
For those that come about through conversion, so that we immediately have also the figures through which each thing will be. It is clear, then, that every proposition is proved through both modes, both through the impossible and ostensively, and it is not possible for the one to be separated from the other.
CHAPTER XV.
On reasoning from opposites.
In which figure it is possible to syllogize from opposite propositions, and in which it is not, will be made clear as follows. I say that opposite propositions are, according to wording, four: as "to all" and "to none," "to all" and "not to all," "to some" and "to none," and "to some" and "not to some"; but according to truth, three, for "to some" and "not to some" are opposed only according to wording; of these, the contraries are the universals, "to all" and "to none," as "every discipline is studious," "no discipline is studious," and the others are opposites. In the first figure, then, there is no syllogism from opposite propositions, neither affirmative nor negative; affirmative indeed, because it is necessary that both propositions be affirmative, but affirmation and negation are opposites; negative indeed, because opposites predicate and deny the same thing of the same thing, but in the third figure the middle is not said of both, but something else is denied of the one, and the same is predicated of the other, and these are not opposed.
CAPUT XVI.
De ratiocinatione ex oppositis in secunda figura,
In media autem figura.et ex oppositis, et ex contrariis contingit fieri sylJogismum.Sit enim bonum quidem in quo a, disciplina autem in quo et c; si ergo omnem disciplinam studiosam sumpsit,et nulIam,a inest omni b,et nulli c,quare nulli c, nulla ergo disciplina disciplina est. Similiter autem et si omnem sumens studiosam disciplinam,medicinam vero non studiosam sumpsit,nam a bquidem omni, autem nulli,quare aliqua disciplina non erit discipIina.Et si a quidem omni,b autem nulli, est autem quidem disciplina, autem medicina,a vero opinio,nulIam enim disciplinam opinionem sumens sumpsit aliquam disciplinam esseopinionem.Differt autem a priore in terminis converti,nam prius quidem ad b, nunc autem rd affirmativum. EL si sit non universalis altera propositio,similiter,semper enim medium est, quod ab altero quidem negative dicitur, de altero vero affirmative. Quare contingit opposita quidem perfici, non autem semper,neque omnino,sed sic se habeant, quae sunt sub medio, ut vel eadem sint, vel totum ad partem; aliter autem impossibile, non enimerunt propositiones ullo modo, neque contrariæ, neque oppositæ.
CHAPTER XVI.
On reasoning from opposites in the second figure.
In the middle figure, syllogisms can arise both from opposites and from contraries. For let good be that in which A is, and let knowledge be that in which B and C are. If, then, he assumed that every branch of knowledge is good, and that no branch of knowledge is good, then A belongs to every B and to no C; wherefore to no C. Hence no branch of knowledge is a branch of knowledge. Likewise also if, assuming every branch of knowledge to be good, he assumed medicine not to be good; for A belongs to every B, but to no C; wherefore some branch of knowledge will not be a branch of knowledge. And if A belongs to every B and to no C, and B is knowledge while C is medicine, and A is opinion—for assuming no branch of knowledge to be opinion, he assumed some branch of knowledge to be opinion. But it differs from the previous case in the conversion of terms; for previously it was in relation to B, but now in relation to C, the affirmative proposition. And if one of the propositions is not universal, it works the same way; for the middle term is always that which is predicated negatively of one and affirmatively of the other. Wherefore it is possible for opposites to be brought to conclusion, but not always, nor in every case, but only when the things under the middle term are so related as to be either the same or related as whole to part; otherwise it is impossible, for the propositions will in no way be either contraries or opposites.
CAPUT XVII.
De syllogismo ex oppositis ira tertia figura.
Intertiaverofiguraaffirmativusquidem sytlogismus nunquam eritex oppositis propositionibus propter causam dictam et in prima figura.Negativus autem erit sylJogismus,et universalibus,et non universafibus terminis. Sit enim disciplina in quo et c, medicina autem in quo a; si ergo sumat omnem mecHcinam, disciplinam, et nuHem medicinam discipIibam,b omni a sumpsit,et nulli a,quare erit aliqua disciplina non disciplina. Similiter autem eL si non universaliter sumpta sit a propositio,nam si est aliqua medicina disciplina,et rursum nulla mcdicina discipIina,accidit disciplinam aliquam non esse disciplinam.Suntautemuniversaliter quidem sumptis terminis contrariae propositiones,si auLem particularis altera sit, oppositae, Oportet autem scire quoniarn contingit oppositasic sumere quemadmodum dixim us,omnem disciplinam studiosam esse,et rursum nullam aut aliquam non esse studiosam, quod non solet latere;erit autem per alias interrogationes syllogizare alteram, Rt quemadmodum in Topicis dictum est, sumere.Quoniarn autem affimationum popositiones sunt tres, sexies accidit opposita sumere, aut omni et nulli, aut omni et non omni, aut alicui et nulli, et hoc converti in terminis, ut a omni et nulli c, aut omni et nulli b.
CHAPTER XVII.
On the syllogism from opposites in the third figure.
In the third figure there will never be an affirmative syllogism from opposite propositions, for the reason stated also in the first figure. But a negative syllogism will arise, both with universal and with non-universal terms. For let discipline be that in which C is, and medicine that in which A is; if then one assumes that every medicine is a discipline, and that no medicine is a discipline, one has assumed B of every A and of no A, so that there will be some discipline that is not a discipline. Likewise also if the proposition is not taken universally; for if some medicine is a discipline, and again no medicine is a discipline, it follows that some discipline is not a discipline. Now when the terms are taken universally, the propositions are contraries; but if one of them is particular, they are opposites. It must be known, however, that it is possible to take opposites in the way we have said—that every discipline is studious, and again that no discipline, or some discipline, is not studious—which is not wont to escape notice; but one will syllogize the other through other interrogations, and take it as has been said in the Topics. Since, moreover, there are three affirmative propositions, it happens six times that opposites are taken: either every and none, or every and not every, or some and none, and this by conversion in the terms, as A of every and of no C, or of every and of no B.
aut huic quidem omni, illi vero non omni, et rursum boc converti secundum terminos: similiter autem et in tertia figura.Quare manifestum est de qnoties et in quibus figuris contingit per oppositas propositiones fieri syllogismum.Manifestum est quoniam ex falsis est verum syllogizare, quemadmodum dictum est prius; ex oppositis autem non est, semper enim contrarius syllogismus fit rei (ut si est bonum non esse bonus, aut si animal non animal) eo quod ex contradictione est syllogismus, et subjecti termini aut iidem sunt, aut hic quidem totum, ille autem pars. Palam autem quoniam in paralogismis nihil prohibet fieri hypotbeseos contradictionem,ut si est impar non esse impar, nam ex oppositis propositionibus contrarius erit syllogismus; si ergo sumpserit hoc rnodo,hypotbeseos erit contradictio.
Or if [the predicate belongs] to every one of this [subject], but not to every one of that, and again this [term] is converted according to the terms; and similarly also in the third figure. Hence it is clear in how many cases and in which figures a syllogism can come about through opposite propositions. It is clear that it is possible to syllogize what is true from false premises, as was said earlier; but from opposites it is not possible, for the contrary syllogism is always produced of the thing (for example, if the good is not good, or if the animal is not an animal), because the syllogism arises from a contradiction, and the terms of the subject are either the same, or one is the whole and the other a part. But it is evident that in fallacies nothing prevents a contradiction of the hypothesis from occurring, for example, if the odd is not odd; for from opposite propositions the syllogism will be contrary. If, then, one has assumed it in this way, there will be a contradiction of the hypothesis.
Oportet autem considerare quoniam sic quidem non est contraria concludere ex uno syllogismo (ut sit condusio, quoniam non est bonum, bonum aut aliud quiddam tale),nisi statim hujusmodipropositio surnatur, ut omne animal esse album et non album, hominem autem animal, sed vel assumere oportet contradictionern,ut quoniam omnis disciplina opinio et non opinio, deinde sumere quoniam medicina discipJina.quidcm est, nulla autem opinio, quemadmodum redargutiones fiunt, vel ex duobus syllogismis.Quare esse quidem contraria secundum veritatem quae sumpta sunt,non est alio modo quam hoc quemadmodum dictum est prius.
CAPUT XVIII.
De petitione principii..
In principio autem petere et accipere et quidem, ul in genere,sumere in eo quod non est demonstrare propositum.Hoc autem acciditmultipliciter, nam et si omnino non syllogizatur,et si per ignotiora aut similiter ignota, et si per posteriora quod prius est, demonstratio enim ex prioribus et notioribus est. Horum ergo nullum est petere quod ex principio est, sed quia haec quidem nata sunt per se cognosci, illa vero per alia (nam principia quidem per se, quae autem sub principiis, per alia), quando non per se notum est, per se aliquis conaturoslendere, tunc petit quod ex principio est.
One must consider that it is not possible to conclude contraries from a single syllogism in this way (so that the conclusion is, for instance, that it is not good, or good, or some other such thing), unless one immediately takes a proposition of this sort, such as that every animal is white and not white, and that man is an animal; but one must either assume the contradiction, as that every discipline is opinion and not opinion, and then assume that medicine is a discipline, but no opinion, in the way refutations are produced, or [one must do it] from two syllogisms. Wherefore for the things assumed to be contraries in truth, this is not possible in any other way than as was said before.
CHAPTER XVIII
On begging the principle.
To beg the principle and to take it, and indeed, generally speaking, to assume [what is needed] for what is not being demonstrated, is the proposed [fallacy]. This happens in many ways, for [it happens] both if no syllogism is produced at all, and if [the conclusion is reached] through things less known or equally unknown, and if through things posterior [what is] prior; for demonstration is from things prior and better known. None of these, then, is begging what is from the principle, but since some things are naturally known through themselves, while others through other things (for principles are through themselves, but the things under the principles are through other things), when something not known through itself is attempted to be shown through itself, then one begs what is from the principle.
Hoc autem est sic facere quidem at statam postulet id quod propositum est: contingit autem et transgredientes et ad alia eorum quæ nata sunt per illa ostendi per hæc monstrare quod ex principio est,ul si a ostendatur per b, et perr., autem natum sit ostendi per a, accidit enim idem It per se demonstrare eos qui sic syllogizant, quod faciunt qui parallelas arbitrantur scribere,latent enim ipsi seipsos talia sumentes quae non valent demonstrare,curn non sint paralleIæ.
To do this in this way is just as the proposed goal demands; but it also happens that those who go beyond and demonstrate other things among those which are naturally shown through those [premises] show the principle through these, as if a is shown through b, and b is naturally shown through a; for it happens that those who syllogize in this way demonstrate the same thing through itself, which is what those do who think they are drawing parallel lines; for they themselves escape their own notice by assuming such things as cannot be demonstrated, since they are not parallel lines.
Quare accidit sic syllogizantibus unumquodque esse dicere si est unumquodque, sic autem omne erit perse notum, quod est impossibile.Si ergo aliquis dubitat assuropto dubio quoniam ainest c,similiter et quoniam b, petat autem inesse b,nondurn manifestum si quod in principio est petat,sed quoniam non demonstravit maHHestum,non enim est principum demonstrationis,quod similiter est incertum.Si autem ad sic se habet ut idem sit,aut manifestum quod convertuntur, autinestalterum alteri, quodinprincipio est petit,nam et quoniam a inest b,per illa monstrabit si convertantur, nunc autem hoc prohibet,sed non modus.Si autem hoc faciat,quod dictum est faciet, et convertet per tria, similiter autem et si sumat inesse c, quod similiter incertum sit, ut et si a inest c,nondum quod ex principio petit, sed neque demonstrat.Si autem idem sit a et b, aut eo quod convertuntur,aut eo quod a sequitur ei quod est b, quod ex principio est petit propter eamdem causam, nam fx principio quod valet, prius dictum est a nobis, quoniam per se monstrabitur quod non est per se manifestu:n.Si ergo est in principio petere per se monstrare quod non per se est manifestum, hoc autem et non ostendere quando similiter dubitantur quod monstratur et per quod monstratur, vel eo quod eadem idem,vel eo quod idem eisdem inesse sumitur,in media quidem figura et tertia utrorumque continget similiter quod est in principio
Therefore it happens that those who syllogize in this way say that each thing is, if each thing is; but in this way everything will be known per se, which is impossible. If, then, someone doubts, when something doubtful has been assumed, that c is present, and similarly that b is present, and he asks that b be present, it is not yet clear whether he is begging the principle; but it is clear that he has not demonstrated it, for it is not a principle of demonstration that which is similarly uncertain. If, however, matters stand such that it is the same thing, either it is manifest that they are convertible, or one is present in the other, he begs the principle; for he will show that a is present in b through those things, if they are convertible; but now this prevents it, yet not the manner. If, however, he does this, he will do what has been said, and he will convert through three terms; and similarly also if he assumes that c is present, which would be similarly uncertain, as also if a is present in c, he does not yet beg the principle, but neither does he demonstrate. If, however, a and b are the same, either by the fact that they are convertible, or by the fact that a follows b, he begs the principle for the same reason; for we have said earlier that what is from a principle is such that it will be shown per se that which is not manifest per se. If, therefore, to beg the principle is to show per se that which is not manifest per se, this is also not to show when both what is demonstrated and that through which it is demonstrated are similarly doubted, either because the same things are the same, or because it is assumed that the same things are present in the same things; in the middle and third figures it will happen similarly for both, which is [to beg] the principle.
petere, in praedicativo quidem syllogismo et in tertia figura, et in prima, negative autem quando eadem ab eodem, et non similiter utraeque propositiones,similiterautem et in media, eo quod non convertuntur termini secundum negativos sylJogismus.Est autem in principio petere in demonstrationibus quidem quæ secundumveritatem sic se habent,in dialecticis autem, quae secundum opi nionem.
CAPUT XIX.
De non propter hoc accidere falsum.
Xon propter hoe autem accidere Falsum (quod saepe in disputationibus solemus dicere) primum quidem est in iis qui ad impossibile syllogismis, quando ad contradictionem esthujus quod monstratum est ea quae ad impossibile. Nam neque qui non contradicit dicit non propter boc,sed quoniam falsum est aliquid poaitum priorum, neque in osten siva, non enim ponit quod contradicit.Amplius autem quando interimitur aliquid ostensive per a c,non est dicere quoniam non propter quod positum est factus est syllogismus,nam non propter hoc fieri tunc dicimus, quandointerempto hocnihilominus perficitur syIlogismus,quod non est in ostensivis, interempta enim propositione, nec qui ad hanc est erit syllogismus. Manifestum igitur quoniam'in iis qui ad impossibile sunt dicitur non propter hoc,et quando sic se habet ad impossibile quæ ex principio est hypothesis.
To beg the question occurs in the predicative syllogism both in the third figure and in the first; in the negative case, however, when the same thing is predicated of the same thing, and not both propositions alike; and likewise in the middle figure, because the terms are not converted according to negative syllogisms. Now to beg the principle occurs in demonstrations that stand thus according to truth, but in dialectical arguments according to opinion.
CHAPTER XIX.
On the false coming about not on account of this.
The false coming about not on account of this (which we are accustomed often to say in disputations) occurs first of all in those syllogisms that lead to the impossible, when the things that lead to the impossible stand in contradiction to that which has been demonstrated. For neither does he who does not contradict say "not on account of this," but [says] that something of the prior premises has been falsely assumed; nor [does this apply] in the ostensive [syllogism], for it does not assume what contradicts. Moreover, when something is refuted ostensively through A C, it cannot be said that the syllogism was made not on account of what was assumed; for we say that it comes about not on account of this when, this being refuted, the syllogism is nonetheless completed—which is not the case in ostensive syllogisms; for when the proposition is refuted, neither will the syllogism that depends on it exist. It is clear, therefore, that "not on account of this" is said in those [arguments] that lead to the impossible, and when the hypothesis from the principle stands thus in relation to the impossible.
ut cum sit,veI cum non sit haec, nihilominus accidit impossibile Ergo manifestissimusquidem modus est non propter suppositionem esse faIsum,quando ab hypothesi inconjunctus est a mediis syllogismus ad irnpossibile,quod dictum estin Topicis, quod enim non est causa, ut causam ponere hoc est; ut si volens ostendere quoniam asymeter est diameter,conetur Zenonis ratione quoniam non est moveri, et ad boc inducat impossibile, nullo enim modo coatinuum est falsum, locutioni quae est ex principio.
as when this is or is not, the impossible nevertheless follows. So the most manifest mode of the false conclusion not arising from a supposition is when the syllogism leading to the impossible is disconnected from the middle terms of the hypothesis, as was said in the Topics; for to posit as a cause what is not a cause is this: for example, if someone wishing to show that the diagonal is incommensurate should attempt by Zeno's argument to show that it does not move, and bring in the impossible for this purpose; for in no way is the false connected with the statement from the beginning.
Alius autem modus,si continuum quidem sit impossibile hypothesi, non tamen propter illam accidat, hoc autem possibile est fieri, et in hoc quod superius,et in hoc quod inferius sumenti, continuum, ut si a ponatur inesse b, autem c, vero d, hoc autem sit falsum inesse d, nam (si ablato a,nihilominus inest c, et d) non erit falsum propter eam quae ex principio est hypothesin.Aut rursum si quis in superiori sumat continuum, ut si a quidem b, autema, /' vero e, falsum autem sit finessea, nam et sic nihilominus erit impossibiIe,interempta quae est ex principio hypothesi.Sed oportet ad eos qui ex principio terminos copulare impossibile, sic enim erit propter hypothesin, ut in inferiori quidem sumenti continuum ad praedicatum terminum; nam si impossibile est a inesse d,interempto a, non amplius erit faIsum.In superiori autem de quo praedicatur; nam si non possibile est inesse b, interempto non amplius erit impossibile; similiter autem et cum privativi sint syllogismi. Manifestum ergo quoniam cum impossibile non ad priores terrninos,non propter positionem accidit falsum; an nec si semper propter hypothes?s erit falsum?
But there is another mode: if the impossible is indeed continuous with the hypothesis, yet does not occur on account of it—this can happen both when one takes the continuous term in what is above and in what is below, as if A is posited to be in B, and C in A, and D in C, and let it be false that D is in A; for if A is removed, C is nonetheless in D, and it will not be false on account of the hypothesis made at the beginning. Or again, if someone takes the continuous term in the higher position, as if A is in B, and B in C, and let it be false that A is in C; for in this way too it will nonetheless be impossible once the hypothesis made at the beginning is destroyed. But one must join the impossible to the terms from the beginning; for thus it will be on account of the hypothesis, as when one takes the continuous term in the lower position up to the predicate term; for if it is impossible for A to be in D, once A is removed it will no longer be false. In the higher position, however, [one must join it] to that of which it is predicated; for if it is not possible for A to be in B, once it is removed it will no longer be impossible; and similarly also when the syllogisms are privative. It is evident, therefore, that when the impossible is not [joined] to the prior terms, the false conclusion does not occur on account of the positing; or will it not even so always be false on account of the hypothesis?
nam si non ei quod est b, sed ei quod esl positum est inesse a, Ic autem c, et hoc d, etsic rnanetimpossibile; similiter autem etin sursum sumenti terminos, quare (quoniam cum est, et cum non est, hoc accidit impossibile) non erit propter positionem,aut cum non est hoc, nihilominus fieri falsum. Nee sic sumendum ut alio posito accidat impossibile, sed quando ablato hoc idem per reliquas propositiones concluditur impossibile, eo quod idem falsum accidere per plures hypotheses nihil fortass inconveniens est, ut parallelas, contingere, et si major est qui interius est, eo qui exteri us, et si triangulus habet plures rectos duobus.
CAPUT XX.
De falsa ratiocinatione, catasyllogismo, hoc est corra-
tiocinatione, et elencho.
Falsa autem oratio fit propterprimum falsum: aut enim ex duabus propositionibus aut ex pluribus omnis est Byllogismus; ergo si ex duabus quidem, harum necesse est alteram, aut etiam utrasqucesse falsas, nam ex veris non erat falsus syIlogismus;si vero ex pluribus (ut sic quidem per a b, boc autem perd y g), horum erit aliquid superiorum faIsum,et propter boc oratio, nam a et per illa concluduntur quare propter illorum aliquid, accidit conclusio et falsum.
For if it is not to B but to the term posited as A that A is taken to belong, and C to B, and this to D, and so the impossible remains; and similarly also for one taking the terms upward, so that (since both when it is and when it is not, this impossible results) it will not be on account of the hypothesis, or when this is not, the false will nonetheless come about. Nor must one take it thus, that when something else is posited the impossible results, but when this is removed the same impossible is concluded through the remaining propositions, because for the same falsehood to result through several hypotheses is perhaps nothing strange, as for parallel lines to meet, and for the term that is within to be greater than the one outside, and for a triangle to have more than two right angles.
CHAPTER XX.
On false reasoning, catasyllogism, that is, counter-reasoning, and elenchus.
False reasoning comes about on account of the first falsehood; for every syllogism is from two propositions or from more. Therefore if from two, one of these must be false, or even both; for from true premises there would not be a false syllogism. But if from more (as, for instance, in this way through A B, and this through D E F), something of these higher premises will be false, and on account of this the reasoning is false; for A and the rest are concluded through those, so on account of something of those, the conclusion and the falsehood result.
Ut autem non catasyllogizelur,observandum quando sine conclusionibus interrogat orationem, ut non detur his idem in propositionibus,eo quod scimus quoniam sine medio syllogismus nun fit, medium autem est quod plerumque dicitur. Quomodo autem oportet ad unanquanque conclusionem observare medium manifestum est, eo quod sciturquatein unaquaque figura ostenditur,hoc autem nos non latebit,eo quod videmus quomodo subrnittimusoratio. nero. Oportet autem quod custodire praecipimus respondentes, ipsos argumentantes tentare latere,boc autem erit primum quidem si conclusiones non prius syllogizent, sed sumptis necessariis non manifestae sint. Amplius autem si non propinqualinterrogant,sed quam maxime longe media,ut sisitopportunum conc:uderea e {, media e de, oportetergo inquirere si a b, et rur:.um non si e sed si e, deinde si c, et sicreliqua, et si per unum medium sit syllogismus, a medio incipere, maxime enim sic latebit respondentem.
But in order that he may not be syllogizing falsely, it must be observed when he questions the argument without conclusions, that the same thing is not granted to him in the premises, since we know that a syllogism does not come about without a middle term, and the middle term is what is generally stated. How one ought to guard against each conclusion is clear from the fact that it is known how it is shown in each figure; and this will not escape us, since we see how we set down the argument. But what we have instructed the respondents to guard against, those who are arguing ought to try to conceal; and this will happen first, indeed, if they do not syllogize the conclusions beforehand, but, with the necessary premises taken, they are not evident. Furthermore, if they do not ask the questions closely connected, but as far as possible at a distance from the middle terms—so that, if it is expedient to conclude that A is in E, the middle terms are D, E—one ought therefore to inquire whether A is in B, and again not whether it is in E, but whether it is in D, then whether it is in C, and so on for the rest; and if the syllogism is through one middle term, to begin from the middle term; for in this way especially will it escape the respondent.
Quoniam ergo habemus quando et quomodo se habentibus terminis fit syilogismus, manifestum at quando erit, et quando non eritelenchus, nam omnibus affirmativis, vel permutatim positis responsionibus (ut bac quidem affirma,tiva,illa vero negativa), contingit fieri elenchum: erit enim syllogismus, et sic in illo modo se habentibus terminis; quare si id quod positum est contrarium sit conclusioni, necesse est fieri elenchum, nam elenchus syllogismus contradictionis est. Si vero nihil affirmetur, impossibile est fieri elenchum, nonenim erat syllogismus,cucQomnestermini erant privativi, quare nec elenchus: namsi eIenchus,necesse est syllogismum esse;cum autem estsy!Iogismus, non necesse estelenchum esse. Similiterautem si nihil positum sit secundum responsionem universa liter; nam eadem erit definitio syllogismi et elenchi.
CAPUT XXI.
Since, therefore, we have [established] when and how a syllogism comes about in the case of terms so disposed, it is manifest also when an elenchus will be and when it will not be. For when all the responses are affirmative, or are set down alternately (as when one is affirmative and the other negative), an elenchus can come about; for there will be a syllogism also when the terms are disposed in that way. Wherefore if what has been granted is contrary to the conclusion, an elenchus must come about, for an elenchus is a syllogism of contradiction. But if nothing is affirmed, it is impossible for an elenchus to come about, for there was no syllogism when all the terms were privative; wherefore neither [will there be] an elenchus. For if there is an elenchus, there must be a syllogism; but when there is a syllogism, it is not necessary that there be an elenchus. Likewise, moreover, if nothing has been granted in response universally; for the definition of syllogism and elenchus will be the same.
Chapter XXI
De fallacia secundum opinionem.
Acciditautem quandoque (quemad modum in positione terminorumfallebamur) et secundum opinionem fieri fallaciam, ut si contingal idem pluribus principaliter inesse,et hoc quidem latere aliquern,et putare nulli inesse,iJlud autem scire, ut insit a et per se, et haec omni similiter. Si igitur quidem putet omni a inesse, et hoc d, autem nulli a, et Loc omni d,ejusdem secundum idem habet disciplinam et ignorantiarn.Rursum si quis fallatur circa ea quæ suntex eadem conjugatione,ut si a inest b,hocautem c, et d, opinetur autem a inesse omni b,et rursum nulli o. Simul enim sciet,et non opinabitur inesse; ergo nibil aliud existimat ex iis quam scit,hoc non opinari, scit enim aliquo modo quoniam a inest per b, velut in universal; hoc quod est particulare; quare quod aliquo modo scil, hoc omnino existimat non opinari,quod est impossibile.
On Fallacy According to Opinion.
It happens sometimes (just as we were deceived in the positing of terms) that a fallacy also arises according to opinion: for instance, if it happens that the same thing belongs primarily to several things, and one person is unaware of this and thinks it belongs to none, while another knows that it belongs—say, that A belongs per se, and similarly to every one. If, then, someone thinks that A belongs to every B, and that D belongs to no A, and that A belongs to every D, he has both knowledge and ignorance of the same thing in the same respect. Again, if someone is deceived about things that are from the same conjugation—for instance, if A belongs to B, and B to C and D, but he opines that A belongs to every B and again to no C. For at the same time he will know and will not opine that it belongs; therefore he thinks nothing other than what he knows from these things—namely, that he does not opine this; for he knows in some way that A belongs through B, as the particular in the universal. Wherefore what he knows in some way, he altogether thinks he does not opine, which is impossible.
In eo autem quod prius dictum est, si non ex eadem conjugatione sit medium, secundum utrumque quidem mediorum ambas propositioned non possibile est opinari ut a quidem omni,c autem nulli,bæc autem utraque omni d; accidit autem aut simpliciter aut in aliquo contrariam sumere primam propositionem.Si enim cui inest omnia opiuatur inesse, autem novit, et quoniam a novit, quare si rursum cui nulli.: putat a inesse, cai alicui inest, huic non putat a inesse, quod autem omni putat cui b, rursum alicui non putare cui b, aut simpliciter, aut in aliquo contrarium et; sic ergo non contingit opinari. Secundum utrumque autem unam,aut secundum al: terum utrasque,nihil prohibet a omni b, et d, et rursumcc nulli c.Nam similis hujusmodi I'allacia,veluti fallimur circa particularia, ut si a omni inest, autem omni c,a omni ineritvsi ergo aliquis novit quoniam a cui inestomni,novit etquoniam ei quod estc;sed nihil prohibet ignorare quoniam est,ut si a quidem duo recti,in quo autem triangulus,in quo vero sensibilis trianguIus; opinabitur enim aliquis non esse c,siens quoniam omnis triangulus habet duos rectos: quare simul sciet et ignorabit idem, nam scire omnem triangulum quoniam duobus rectis non simpIexest,sed hoc quidem universalemhabetdisciplinax,illud vero singularem. Sic ergo in universali novit c,quoniam duobus rectis in singulari autem non novit, quare non habebit contrarias.
In what was said before, if the middle term is not from the same combination, it is not possible to hold both propositions according to each of the middle terms—for instance, that A belongs to every, C to none, and both of these to every D. But it happens that one takes the first proposition as contrary either simply or in some respect. For if someone opines that A belongs to everything to which it belongs, and knows that it belongs, and knows that it belongs to C, then if again he thinks that A belongs to nothing to which C belongs, he does not think that A belongs to that to which it does belong; but what he thinks belongs to everything to which B belongs, again he does not think belongs to something to which B belongs—either simply or in some respect it is contrary. Thus, then, it is not possible to opine. But according to both one proposition, or according to one both propositions, nothing prevents A belonging to every B and D, and again C to no C. For the fallacy of this sort is similar, as we are deceived about particulars—for instance, if A belongs to every B, and B to every C, A will belong to every C. If, therefore, someone knows that A belongs to everything to which it belongs, he also knows that it belongs to C; but nothing prevents him from being ignorant that C exists—for instance, if A is two right angles, and that in which it is is a triangle, and that in which it is is a perceptible triangle. For someone will opine that C does not exist, knowing that every triangle has two right angles; therefore he will at the same time know and be ignorant of the same thing. For to know that every triangle has two right angles is not simple, but the one has universal knowledge, the other singular. Thus, then, in the universal he knows C, that it has two right angles, but in the singular he does not know it; therefore he will not have contrary opinions.
Similiter autem est quae in Menone est oratio,quoniam disciplina est reminiscentio; nunquam enim accidit praescire quod singulare est, sed simul inductione sumere particularium disciplinam, velut recognoscentes. Nam quaedam scientes,statim scimus, ut quoniam duobus rectis, si scimus quoniam triangulus, similtterautemet in aliis. Ergo universali quidem speculamur particularia, propria autem non scimus; quare contingit et falli circa ea, verum non contrarie, sed habere quidem universale, decipi autem particulari. Similiter autem in praedictis non enim contraria quæ est secundum medium ei quae est secundum syllogismumdisciplinæ,nec quae est se. cundum utrumque mediorum opinatio,nihilenim pro. hibet scientem, et quoniam atoti inest, et rursum hoc totic,putare non inesse, ut quoniam omnis mula sterilis, et hæc mula, putare hanc habere in utero; non enim scit quoniam a, qui non conspicit,quod est secundum utrumque.Quare manifestum quoniam et si boc quidem novit, illud vero non novit,falletur, qdod habent universales ad particulares disciplinas; nullum enim sensibilium extra sensum fit scimus, nee si sentientes fuerimus sciæus,nisi etinuniversali,el in eo quod habet propriam disci pIinam,sed non in eo quod estin actum.Nam scire tripliciter dici. tur,aut ut universali.
Similarly, the argument in the Meno—that learning is recollection—is of this sort; for it never happens that one foreknows the particular, but one grasps the knowledge of particulars together with the induction, as it were recognizing them. For there are things which, knowing, we know at once—for instance, that something is equal to two right angles, if we know that it is a triangle; and similarly in other cases. So by the universal we contemplate particulars, but we do not know the proper characteristics; hence it happens that we are also deceived about them—not, however, in a contrary way, but we have the universal and are deceived in the particular. Similarly in the cases mentioned before; for the opinion according to the middle term is not contrary to the knowledge according to the syllogism, nor is the opinion according to both middle terms. For nothing prevents one who knows both that A belongs to the whole and again that this whole belongs to something, from thinking that it does not belong—for instance, that every mule is sterile, and that this is a mule, yet thinking that this one has a fetus in the womb; for he does not know the A which he does not observe, namely that which is according to both. Wherefore it is clear that even if he knows one thing but does not know the other, he will be deceived—which is what universals have in relation to the knowledge of particulars. For we know none of sensible things apart from sense, nor, if we have perceived, do we know, unless in the universal and in that which has the proper knowledge, but not in that which is in act. For to know is said in three ways: either as in the universal...
aut ut propria, aut ut in actu quare et decipi totidem modis, nihil ergo prohibete scire,et deceptum esse circa idem, verumtamen non contrarie.Quod accidit et ei qui secundum utramque scit propositionuu',et non pertractavit prius,nam opinans in utero habere mulam, non habet secundum actum disciplinam,neque propter opinionem fallaciam contrariam disciplinæ,syllogismus enim est contraria fallacia in universali.Qui autem opinatur quod bonum esse est malum esse,idem opinabitur bonum esse et malum. Sit enim bonum esse in quo a,malum autem essein quo b,rursurn bonum esse in quo; quoniam igitur idem opinatur et etcet esse opinabitur,et rursum esse asimiliter,q uare et e a, nam quemad' modum si erat verum de quo b, et de quo a. et de quo a verum erat,gic et in opinatione Similiter autem et in eo quod est esse.Nam cum idem sit et b, et rursum et a, a idem erit, quare et opinatione similiter; ergo hoc quidem necessarium si quis det primum. Sed fortasse illud falsum opinari aliquem quod malum esse est bonum esse, nisi secundum accidens; multipliciter enim possibile est hoc opinari, perspiciendum autem hoc melius.
Or as proper, or as in act; hence also deception occurs in just as many ways. Nothing therefore prevents one from knowing and being deceived about the same thing, though not in a contrary manner. This happens also to one who knows both propositions and has not examined them beforehand; for one who opines that he has a mule in the womb does not have knowledge in act, nor does he have a falsehood contrary to knowledge on account of the opinion, for the syllogism of the contrary falsehood is in the universal. But one who opines that to be good is to be evil will opine the same thing to be both good and evil. For let "to be good" be in respect of A, and "to be evil" in respect of B; again, "to be good" in respect of A. Since therefore he opines the same thing and also that it is, he will opine it, and again that it is similarly, wherefore also A; for just as if it was true of that of which B, also of that of which A, and of that of which A it was true, so also in opining. And similarly also in respect of being. For since the same is both B and again A, A will be the same, wherefore also in opining similarly; therefore this is indeed necessary if one grants the first. But perhaps that is false—that someone opines that to be evil is to be good—except according to accident; for it is possible to opine this in many ways, but this must be examined better.
CAP'JT XXII.
De conversionibus terminorum.
Quando vero convertuntur extremitates, necesse est et medium converti ad utramque; si enim a de per est, si convertitur et inest cui a omni, et a convertitur, et inestcui a omni,b per medium c, etc convertitur per medium a. Et in non esse itidem, ut si inest c, a vero non inest b, neque a inerit c. Si ergo convertatur ad a, et ad a convertetur: sit enim nulli a inexistens, ergo neque c, omni enim inerat 6, et si convertitur ad c, et a convertetur ad c; nam de quocunqueomnino 6, et c.Et. sic ad a convertitur, et convertetur ad a: cui enim inest,etc; cui autem c, a. non inest; et solum hoc a conclusione incipit,alia autem non similiter, utin prædicativo sy 1iogismo. Rursum si a et convertuntur, et et similiter, omni autem necesse est a aut inesse,et etdsic sehabebunt, utomni alterurn insit; quoniam enim cui a b, et cui d, omni autem a aut c, et non simul, manifestum quoniam et aut omni,et non simuI, utsi ingenitum, incorruptibile,et incorruptibile ingenitum, necesse est quod factum est corruptibile corruptibile factum esse, duo enim:syllogisml constituti sunt. Rursum si omni quidem,a vel b,et cvel d, simul autem non insunt,si convertitur net c, et etd convertetur. Nam si alicui non inest b, cuid,palam quoniama inest; si autem a,et c,convertuntur enim; quare simul et d, hoc autem impossibile.
CHAPTER XXII.
On the Conversions of Terms.
But when the extremes are converted, the middle term must also be converted to both; for if a is present in every b through c, and if it is converted and is present in that to which a belongs universally, then a is converted, and it is present in every thing to which a belongs, b through the middle c, and c is converted through the middle a. And in the case of non-being likewise: as if c is present, but a is not present in b, then neither will a be present in c. If, therefore, it is converted to a, it will also be converted to a; for let it be present in none to which a is present; therefore neither will c be present, for it was present in every b, and if it is converted to c, a will also be converted to c; for of whatever thing b is universally predicated, of that thing c also is. And thus it is converted to a, and it will be converted to a: for that in which it is present, etc.; but that in which c is present, a is not present; and this alone begins from the conclusion, but the others do not similarly, as in the predicative syllogism. Again, if a and b are converted, and c and d similarly, but it is necessary that a or c be present in every thing, and they are so disposed that either is present in every thing; for since that in which a is present, b is present, and that in which c is present, d is present, but a or c is present in every thing, and not simultaneously, it is manifest that b or d is present in every thing, and not simultaneously; as, if the ungenerated is incorruptible, and the incorruptible is ungenerated, it is necessary that what has been made is corruptible and the corruptible has been made; for two syllogisms have been constituted. Again, if a or b is present in every thing, and c or d, but they are not present simultaneously, if b is converted with c, then d will also be converted. For if b is not present in some thing, it is clear that a is present in that thing; but if a is present, then c is present, for they are converted; wherefore d is also present simultaneously, but this is impossible.
Quando autem u toti bet inest, et de nullo alio prædicatur, inest autem et omni c, necesse esta etbconverti, quoniamenim desoJis bcdicitura, prædicaturautem bbet idem dese et de c,manifestum quoniam de qui bus a, et dicetur omnibus, verum et de a. Rursum quando a et b, toti insunt convertitur autem b, necesse est a omni inesse, quoniam enim omni ca, autem b,eo quod convert.untur, et a omni inerit. Quando autem duo fuerint opposita,ut a magis eligendum sit quam b,cum sint opposita, et quam csimiIiter,si magiseligenda sunta cquam d,a magis eligendum quamd. Similiterenim sequendum a,etfugiendum b,opposita enim, et ei quod est d,narn et haec opponuntur; si ergo a ei quod est simililereIigend um,etb ei quod est fugiendum,utrumque enim utrique similiter fugiendum eligendo; quare et hæc ambo a iis quae suntb d,quoniarn autem magis, non possibile similiter,nam et similiter erunt.Si autem dmagis eligendum quam a,et quam cminus fugiendum; nam quod minus est minori opponitur; magis autem eligendum est majus bonum et minus malum quam minus bonum et majus malum.
But when a is present in the whole of b and is predicated of no other, and is also present in every c, it is necessary that b also be converted; for since it is said of every b, and b is predicated of the same things as c, it is manifest that it will be said of all the things of which a is said, and also of a. Again, when a and b are present in the whole, and b is converted, it is necessary that a be present in every c; for since a is in every c, and b is converted, a will also be present in every b. But when two things are opposed, as that a is more to be chosen than b, when they are opposed, and likewise than c, if a and c are more to be chosen than d, then a is more to be chosen than d. For similarly one must pursue a and avoid b, for they are opposites, and likewise in relation to d; for these too are opposed. If, therefore, a is similarly to be chosen and b similarly to be avoided, each of the two is similarly to be avoided and chosen; wherefore both these things are so in relation to the things that are b and d. But since it is more, it is not possible that they be similarly; for they will also be similarly. But if d is more to be chosen than a, and less to be avoided than c; for what is less is opposed to what is less; but more to be chosen is the greater good and the lesser evil than the lesser good and the greater evil.
Universum igitur magis eligendum quam a c, nunc autem non est, ergo magis a eligendum quam d,et ergo minus fugiendum quam b.Si ergo eligat omnis amans secundum amorem a sic sebabere,ut concedere, et non concedere in quo c, aut concedere in quo d, et non tale esse ut concederein quo b,manifestum quoniam a hujusmodi esse, magis eIigendum est quam concedere;ergo diligi quam conventio magis eligendum secundum amorem;magis ergo amorest in amicitia quam convenire. Si antem maxime hujus, et finis bæc, ergo convenire aut non est omnino, aut diligendi gratia,narn et alix concupiscentiæ etartes sic fiunt.Quomodo ergo se habent termini secundum conversiones, et in eo quod magis fugiendum vel magis eligendum -sit, manifestum est.
CAPUT XlIII.
De epagoge, id est inductione.
Quoniam autem non solum dialectici et demonstra. tivi syllogismi per prædialectici fiunt figuras, sed et rhetorici, sed et simplicitea quaecunque fides est,et secundum unamquamque artem,n unc eritdicend um Omnia enim credimus per syllogismum aut ex inductione; ergo si inductio quidem est, ex inductione syllogismus peralterum extremitatem medio syUogizare.Ut si eorum quae sunt a medium sit b,per ostendere a inesse b, sic enim facimus inductiones.
Therefore the whole is more to be chosen than a c, but now it is not so; therefore a is more to be chosen than d, and therefore less to be shunned than b. If, then, every lover chooses according to love—a thus, for example, to concede, and not to concede in the case of c, or to concede in the case of d, and not to be such as to concede in the case of b—it is manifest that a being of this sort is more to be chosen than to concede; therefore to be loved is more to be chosen, according to love, than agreement; therefore love is more in friendship than to agree. But if this is most of all, and these are the end, then to agree is either not at all, or for the sake of being loved; for other desires and arts also come about in this way. In what way, then, terms stand in respect of conversions, and in respect of what is more to be shunned or more to be chosen, is manifest.
CHAPTER XLIII.
On epagoge, that is, induction.
Since, however, not only dialectical and demonstrative syllogisms are produced through the figures by pre-dialectical [arguments], but also rhetorical ones, and indeed simply whatever is a matter of belief, and according to each art, it must now be said. For we believe all things either through syllogism or from induction; therefore if induction indeed is [this], the syllogism from induction [proceeds] by syllogizing through the other extreme with the middle. As, for example, if b is the middle of the things that are a, by showing that a is present in b; for thus we make inductions.
Ut si a longævum, in quo autem choleram non habere, in quo vero singulare longævum,ut homo equus, et muIus.Ergo toti inest a, omne enim quod sibi cholera est,longævum,sed et non habere cho.leram, omni inest c; si ergo convertatur ei quod est b,et non transcendat medium,necesse este inesse, 6. Ostensum enim est prius quoniam,si duo aliqua eidem insunt,et ad alteram eorum convertatur extremum, converso et alterum inerit praedicatorum. Oportet autem intelligere ex singularibus omnibus compositum, nam inductio per omnia. SyIlogismus autem hujusmodi est primae et immediatae propositionis v quarum enim est medium, per medium est syllogismus; quorum vero non est, per inductionem. Et quodam modo opponitur inductio syllogisrno,nam hic quidem per medium extremum de tertio ostendit,illa autem per tertium extremum de medio. Ergo natura quidem prior et notior per medium syllogismus, nobis autem manifestior qui est per inductionem.
For example, if a is long-lived, and that in which there is no bile, and that in which there is the individual long-lived thing, such as man, horse, and mule. Therefore a belongs to the whole; for everything that is free of bile is long-lived, but also not to have bile—c belongs to every such thing. If, then, b is converted to it and does not transcend the middle term, it is necessary that it be present. For it was shown earlier that, if two things are present in the same subject and the extreme is converted to one of them, the other of the predicates will also be present in the converted term. But one must understand the composite from all the singulars; for induction is through all. The syllogism of this kind is of the first and immediate proposition; for those things of which there is a middle term, the syllogism is through the middle term; but those of which there is no middle term, it is through induction. And in a certain way induction is opposed to syllogism; for the latter shows the extreme concerning the third through the middle term, while the former shows the middle through the third extreme. Therefore by nature the syllogism through the middle term is prior and more known, but to us the one that is through induction is more manifest.
CAPUT XXIV.
De paradigmate, hoc est exemplo.
Exemplum autem est, quando medio extremum inesse ostenditur per id quod est simile tertio.Oportet autem et medium tertio,et primum simili notius esse, inesse. Ut sit a malum, autem contra confines inlerre bellum,in quo autem A thenienses contra Thebanos,in quo autem Tbebanos contra Phocenses. Siergovolumus osterderequoniamThebanis pugnare malum est, sumendum quoniam contra confines pugnare est malum,hujus autem fides ex similibus,ut quoniam Thebanis contra Phocenses.Quoniam ergo contra confines malum,contra Thebanos autem contra confines est, manifestum quoniam contra Thebanos pugnare malum.Quonjam ergo beet inest,manifestum,utrumque enim est contra confines inferre bellum,et quoniam a d,Thebanis non enim fuit utile contra Phocenses bellum.Quoniam autem a inest b, per d. osteadetur,eodem autem modo et si per plura similia fides fiat medii ad extremum. Manifestum ergo quoniam exemplum est neque ut totum ad partern,neque ut pars ad totum, sed ut pars ad partem, quandoamboquideminsuntsubeodemnotumautem alterum. Etdiffertabinductione, quoniam hæcquidem ex omnibus indivjduisostendebat inesse estremum rnedio,et ad extremus non copulabat syllogigrnum,hoc autemet copulat,et non ex omnibus ostendit.
CHAPTER XXIV.
On paradigm, that is, example.
An example is when it is shown that the extreme is present in the middle through that which is similar to a third term. It is necessary that both the middle be present in the third and the first be more known than the similar term, and that it be present. For let a be evil, and waging war against neighbors, and that in which the Athenians against the Thebans, and that in which the Thebans against the Phocians. If then we wish to show that for the Thebans to fight is evil, one must assume that to fight against neighbors is evil; the proof of this is from similar cases, as that for the Thebans against the Phocians. Since therefore to fight against neighbors is evil, and against the Thebans is against neighbors, it is clear that for the Thebans to fight is evil. Since therefore b is present in c, it is clear; for both are to wage war against neighbors, and that a is present in d, for the war against the Phocians was not useful for the Thebans. But that a is present in b will be shown through d. In the same way also, if proof of the middle in relation to the extreme is made through several similar cases. It is clear therefore that an example is neither as whole to part, nor as part to whole, but as part to part, when both are present under the same term and one of them is known. And it differs from induction, because induction showed from all individual cases that the extreme is present in the middle and did not join the syllogism to the extreme, whereas this does join it and does not show it from all cases.
CAPUT XXV.
De apagoge deductioneque.
Deductio autem quando medio quidem primum palam est inesse, postremo autem medium dubium quidem similiter autem credibile aut magis conclusione. AmpIius,si pauciora sunt mediapostremoetmedio, omnino enim propinquius esse accidit scientiæ.Ut sit a docibile, in quo disciplina, justitia, ergo disciplina quoniam docibiIis,manifestum; justitia autem si disciplina,dubium.Si if!itur similiter aut magis credibile sit quam a c,deductio est, propinquius enim scientiæ,per quod assumpserint a c, disciplinam prius non babentes.Aut rursum si pauciora media sint c, nam et sic propinquius est scientiæ. Ut si sit quadrangulare,in quo autem e rectilineum,in quo j circulus,si ergo ejus quod est et unum solum sit medium,per lunares figuras æqualem fieri rectilineo circulum propinquius erit scientis. Quando autem neque credibilius est quam a c, neque pauca media, non dico deductionem, neque quando immediata est c, disciplina enim quod hujusmodi est.
CHAPTER XXV.
On reduction and deduction.
Reduction, however, is when it is clear that the first term is present in the middle, but it is doubtful, though similarly or more credible than the conclusion, that the middle is present in the last term. Further, if the middle terms between the last term and the middle are fewer, for it happens to be altogether closer to knowledge. For example, let A be teachable, in which is discipline; let C be justice. That discipline is teachable is manifest; but whether justice is discipline is doubtful. If, then, it is similarly or more credible than the conclusion A C, there is a reduction; for that through which they have assumed A C is closer to knowledge, since they did not previously have discipline. Or again, if the middle terms of C are fewer, for in this way too it is closer to knowledge. For example, if it be a quadrilateral figure, in which is E, a rectilinear figure, in which is F, a circle: if, then, there is only one middle term of that which is E, that the circle is made equal to a rectilinear figure through lunular figures will be closer to knowledge. But when it is neither more credible than the conclusion A C, nor are the middle terms few, I do not call it a reduction, nor when C is immediate; for knowledge is of what is of this sort.
CAPUT XXVI.
De instantia,quam enstasin dicunt.
Instantia autem est propositio propositioni contraria.Dilfert autem a propositione,quoniam contingit quidem instantiam esse in parte, propositionem vero aut omnino non contingit,aut non in universalibus syllogismus. Fertur autem instantia duobus modis et per duas figuras: duobus modis quidem, quoniam aut universalis aut particularis omnis instantia; per duas autem figuras, quoniam oppositae feruntur propositioni, opposita autem in prima et tertia figura perficiuntur soIis.Nam quando postutatur omni inesse, instamus quoniam nulli, aut quoniam alicui non inest. Horum autem nulli quidem ex prima figura, alicui autem non ex postremat Ut sit a unam esse disciplinam, in quo contraria; proponit ergo unam esse contrariorum discipIinam, aut quoniam omnino non est eadem oppositorum instant. Contraria al1lem opposita,quare fit prima llgura; aut quoniam noti et ignoti non una, haec autem tertia.
CHAPTER XXVI.
On objection, which they call enstasis.
An objection is a proposition contrary to a proposition. It differs from a proposition because an objection may be partial, whereas a proposition either cannot be so at all, or the syllogism does not proceed in universals. An objection is brought in two ways and through two figures: in two ways, because every objection is either universal or particular; through two figures, because objections are brought against a proposition, and opposites are established only in the first and third figures. For when it is claimed that something belongs to every, we object that it belongs to none, or that it does not belong to some. Of these, the one that it belongs to none is from the first figure, but that it does not belong to some is from the last. For example, let A be that there is one science, in which case there are contraries; he therefore proposes that there is one science of contraries, or objects that it is not at all the same [science] of opposites. But contraries are opposites, and so the first figure results; or [he objects] that there is not one [science] of the known and the unknown, and this is the third.
Nam secundum tertiam notum et ignotum contraria quidem esse verum, unam autem esse eorum disciplinam, falsum.Rursum in privativa propositione similiter: cum postulat enim non esse contrariorum unam disciplinam, aut quoniam omnium oppositorum, aut quoniam contrariorum aliquorum est eadem discipHna,dicimus, ut sani et ægri, ergo omnium quideru ex prima, aliquorum vero ex tertia figura, Simpliciter autem in umnibus universaliter quidem instantibus, necesse est ad id quod nniversaie est proposito contradictionem dicere (ut. si non unam existimet contrariorum omnium, dicere oppositorum unam; sic autem necessse est primam esse figuram, medium enim fit universale ad hoc quod ex principio); quod autem ad hoc in parte est universale, dicitur propositio, ut noti et ignoti non eadem, nam contraria universale ad bæc, et fit tertia figura, medium enim in parte sumptum, ut notum et ignotum. Nam ex quibus est syllogizare contrarium, ex iis et iastantias conamus dicere, quare et ex his solis figuris ferimus,nam in his solis oppositi syllogismi,per mediam enim figuram non fuit arfirmare.Amplius autem et si sit,oratione indiget pIurima, quæ est per mediam figuram, ut si non concedant a inesseb,eo quod non sequitur hoc c, hoc enim per alias propositiones manifestum; non oportet autem instantiam converti ad alia, sed statim manifestam habere alteram propositionem.
For according to the third figure, it is false that contraries are both known and unknown, and that there is one discipline of them. Again, similarly in a privative proposition: for when he demands that there is not one discipline of contraries, either because it is the same discipline of all opposites, or because it is the same discipline of some contraries, we say, as of the healthy and the sick, that it is of all indeed from the first figure, but of some from the third figure. But simply in all cases, when someone objects universally, it is necessary to state a contradiction to the proposition that is universal (for example, if he thinks there is not one discipline of all contraries, to say that there is one of opposites; and thus it is necessary that the first figure result, for the middle term becomes universal in relation to that which is from the principle); but what is universal in relation to this in part is called a proposition, as that the known and the unknown do not have the same discipline, for the contrary is universal in relation to these, and the third figure results, for the middle term is taken in part, as known and unknown. For from the same premises from which it is possible to syllogize the contrary, we also attempt to state objections; wherefore we also bear them from these figures alone, for in these alone are there opposite syllogisms, for through the middle figure it was not possible to affirm. Moreover, even if it were, it requires a very great deal of argument, which is through the middle figure, as if they do not concede that a is in b, because this c does not follow, for this is made clear through other propositions; but an objection ought not to be converted to other things, but immediately to have the other proposition manifest.
Quapropter et signum ex sola hac figura non est. Perspiciendum autem et de aliis instantiis, ut de iis qua? sunt ex contrario, et simili, et secundum opinionem, et si particularem ex prima, vel privativam ex media possibile est sumere.
CAPUT XXVII.
De eicote, enthymemate. hoc est consentaneo signo, indicio, et
Eicos autem et signum non idem est, sed eicos quidem est propositio probabilis. Quod enim ut in pluribus sciunt sit factum;vel non factum, aut esse vel non esse, hoc est eicos, ut odire invidentes, vel diligere amantes.Signum autem vultesse;propositio demonstrativa, vel necessaria,vel probabilis; nam quo ex;stente est, vel quo facto prius vel posterius res,signum est vel fuissevel esse.Enthymema ergo est syllogismus imperfectus ex eicotibus et signis. Accipitur autem signum tripliciter,quoties et medium in figuris,aut enim ut in prima, aut ut in me. dia,aut ut in tertia: ut ostendere quidem parientem esse, eo quod lac babeat, ex prima figura, medium enim lac habere,in quo a parere b, lac habere mulicr in quo c.Quoniam autem sapientes,studiosi,narn Pittacus est studiosus, per postremam, in quo a studiosum, in quo sapientes, in quo Pittacus. Verum igitur etbde praedicari; sed hocquidem non dicunt quia notum sit,illud vero sumunt.
Wherefore the sign is not from this figure alone. But one must also examine the other objections, as concerning those which are from contraries, and from similitude, and according to opinion, and whether it is possible to take a particular from the first figure, or a privative from the middle.
CHAPTER XXVII.
On the eikos and the enthymeme, that is, the consentaneous sign, the indication, and
The eikos and the sign are not the same, but the eikos is a probable proposition. For that which they know happens as in the majority of cases—either done or not done, or being or not being—this is the eikos, as that the envious hate, or that lovers love. But the sign wishes to be a demonstrative proposition, either necessary or probable; for when that of which it is the sign exists, or when it has been done, the thing is either prior or posterior, and it is a sign either of having been or of being. The enthymeme, therefore, is an imperfect syllogism from eikota and signs. But the sign is taken in three ways, as often as the middle term is in the figures: for either as in the first, or as in the middle, or as in the third. For example, to show that she who has given birth is so, from the fact that she has milk, is from the first figure; for the middle term is having milk, in which a is having given birth, b is having milk, and the woman in which is c. But that the wise are studious—for Pittacus is studious—is through the last figure, in which a is the studious, in which the wise, in which Pittacus. Therefore it is true that b is predicated of c; but this they do not say because it is known, while that they assume.
Peperisse autem quoniam pallida,per mediam figuram vult esse; quoniam enim sequitur parientespallor, sequilur autem et hanc, ostensum esse arbitrantur quoniam peperit. Pallor in quo a, parere in quo b. mulier in quo c, Ergo si una quidem dicatur propositio,signurn fit solum, si autem et altera sumitur, syllogismus.Ut Pittacus liberalis, nam ambitiosi Iiberales,Pittacus autem ambiti05uS.Aut rursus,quoniam sapientes b01Ji, Pittacus autem bonus, sed et sapiens, sic ergo fiunt syllogismi. Verum quidem per primam figuram insolubilis, si verus sit, universalis enim est. Qui autem per postremam, est soIubilis,et si vera sit conclusio,eo quod non universaIis, est in tertia, nee ad rem syllogismus, non enim si Pittacus est studiosus, propter hoc et alios necesse est esse sapienles.Qui vera per mediarnfiguram est,semper et omnino solubilis nunquam enim sylIogismusfit,sic se habentibusterminis.Non enim si quæ peperit paIlida,pa lida autem et hæc,necesse est parere hanc; ergo verum est quidem in omnibus figUflS, differentias autem habent jam dictas. An igitur sic dividendum sign um?horumautem medium indicium surnendum,nam indicium dicunt esse quod scire facit, tale autem maxime medium, an vero quae quidem ab extremitatibus signa dicenda,quæ autem ex medio indicium? probabilissimum enim et maxime veram est quod est perprimam figuram.
That she has given birth, because she is pale, he wants to establish through the middle figure; for since pallor follows upon those who have given birth, and this woman is also pale, they think it has been shown that she has given birth. Let pallor be A, having given birth be B, the woman be C. Therefore, if only one proposition is stated, a sign results; but if the other is also assumed, a syllogism results. As, for example, Pittacus is generous, for the ambitious are generous, and Pittacus is ambitious. Or again, that the wise are good, and Pittacus is good, but also wise; in this way, then, syllogisms are formed. The one through the first figure is indeed irrefutable, if it is true, for it is universal. But the one through the last figure is refutable, even if the conclusion is true, because it is not universal; it is in the third figure, and the syllogism does not bear on the matter, for it is not because Pittacus is studious that others must also be wise. The one through the middle figure is always and altogether refutable, for no syllogism results when the terms are so disposed. For it is not the case that if she who has given birth is pale, and this woman is also pale, this woman must have given birth. Therefore, it is true in all the figures, but they have the differences already stated. Should the sign, then, be divided in this way? But of these, the middle must be taken as the indication, for they call an indication that which makes one know, and such is especially the middle term; or should those from the extremes be called signs, and those from the middle an indication? For what is through the first figure is most probable and most true.
CAPUT XXVIII.
De syllogismo physiognomico.
Naturas autem cognoscere possibile est, si quis concedat simul transmutare corpus etanimam,quæcunque sunt naturales passiones; discens enim aliquis fortasse musicam, transmutavit secundum quid animam,sed non earum quae naturanobisinsunt,bæc est passio, sed ut irae et concupiscentiæ, et naturalium motionurn.Si igitur et hoc det,et unum unius signum esse,et possumus sumere proprium uniuscujusque generis passionem et signum, poterimus naturas cognoscere.Si enim est proprie alicum generi individuo existens passio, ut si leonibus fortitudo, necesse est et signum esse aliquod, compati enirn sibi invicem positum est,et sit hoc magnas summitates habere, quod et aliis generibus,non totis contingit.Nam signum sic propriurn est,quoniam totius generis propria passio est, et non solius proprium, sicut solemus dicere.Erit ergo et in alio genere hoc, eterit fortis homo, et aliquod aliud animal; habebit ergo signum, unum enim unius erat. Si ergo bæc sunt,poterimus talia signa colligere in iis animatibus quae solum unam passionem habent aliquam propriam,unaquæque autem hahet signum,et quoniam unum habere necesse est,poterimus naturas cognoscere.Si vero duo habet propria totum genus, ut leo, forte et communicativum, quomodo cognoscemus utrum utrius sit signum,eorum signorum quae proprie sequuntur?
CHAPTER XXVIII.
On physiognomic syllogism.
It is possible to know natures if one grants that body and soul change together in whatever natural affections there are; for someone learning music has perhaps changed his soul in some respect, but this is not one of the affections that are natural to us, but rather such things as anger and desire, and natural motions. If, then, one grants this too, and that one thing is the sign of one thing, and we can grasp the proper affection of each kind and its sign, we shall be able to know natures. For if there is an affection that belongs properly to some one kind, as strength to lions, there must also be some sign of it; for it is laid down that they are affected in sympathy with one another, and let this be having great extremities, which also happens to other kinds, though not to all. For a sign is proper in this way, because the affection is proper to the whole kind, and not proper to it alone, as we are accustomed to say. Therefore this will also be in another kind, and there will be a strong man, and some other animal; it will therefore have the sign, for one thing was the sign of one thing. If, then, these things are so, we shall be able to collect such signs in those animals that have only one proper affection, and each has a sign, and since it is necessary to have one, we shall be able to know natures. But if the whole kind has two proper affections, as a lion perhaps has both strength and a capacity for sharing, how shall we know which of the signs that properly follow is the sign of which?
An si et alii alicui non toti ambo,et in quibus non lotis utrumque,quando hoc quidem habet,illud autem non?nam sifortis quidem, liberalis autem non,habet autem duorum hoc,palam quoniam et in leone hoc signum fortitudinis.Est vero naluras cognoscere in prima quidem flgura,eo quod medium priori extremitati convertitur, tertiam autem transcendit,et non convertitur,ut sit fortitudo a, summitates magnas habere in quo b, autem leo; ergo cui c, omni, sed et aliis,cui autem b, a omni, et non pluribus, sed; convertitur si autem non, non erit unum unius signum.
Or if both are not present in some other whole, and in those in which both are not present, when one has this, the other does not? For if he is brave but not generous, and he has one of the two, it is clear that this is also a sign of bravery in the lion. But it is possible to know natures in the first figure, because the middle term is converted with the first extreme, and it transcends the third and is not converted, so that bravery is a, having great extremities is in what b is, and the lion is c; therefore it belongs to every c, but also to others; and a belongs to every b, and not to more, but; if however it is not converted, there will not be one sign of one thing.